1. Introduction

Phase-sensitive amplifiers (PSAs) offer several advantages over phase-insensitive amplifiers (PIAs) like the erbium-doped fiber amplifier (EDFA); they exhibit a theoretical 0 dB noise figure (NF) [1], can mitigate transmission nonlinearities [2,3] and are wavelength flexible with a potentially large gain bandwidth [4]. These appealing aspects of PSAs make them attractive for applications such as pre-amplification in free-space [5] and fiber-optical transmission links [2,6], all-optical phase and amplitude regeneration [7] and all-optical sampling [8,9].

PSAs, like other parametric amplifiers, rely on nonlinear interaction in either a three- [10] or four-wave mixing (FWM) process [11] where one or several pump waves exchange energy to weaker signal and idler waves. A common and practical setup for a PSA is the two-mode, degenerate pump configuration [4], where signal and idler waves are equally spaced apart in frequency around a central degenerate pump wave, see Fig. 1(a). The relative phase between these three waves determines the exchange of energy between them. The gain of such an amplifier can be written as [4]

Here, $|\mu |^2=G_\textrm {PIA}$ is the PIA gain the signal wave experiences should the idler be absent at the amplifier input, $|\nu |^2=G_\textrm {PIA}-1$, $\phi _{s,i,p}$ is the phase of the signal, idler, and pump waves respectively and $\phi _\nu$ is a constant phase.

 

Fig. 1. a) PSA with input signal (of power $P_s$), idler, and pump waves. b) 50/50-coupler with input powers $P_\textrm {in,1}$ and $P_\textrm {in,2}$ and output powers $P_1$ and $P_2$.

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For the PSA to achieve phase-sensitive gain and reach the attractive $<3$ dB NF (lowest measured fiber-PSA NF: 1.1 dB [12]) the three waves need to be locked in phase at the PSA input. In the case of an optical link, while correlated signal, idler and pump waves are commonly produced in the transmitter (Tx), usually by employing a copier scheme [4], it is practically difficult to create a correlated receiver (Rx)-side, high- power and quality, pump wave.

In previous PSA demonstrations, the pump was regenerated for the PSA, usually by co-propagating the Tx pump in the link along with the signal and idler for frequency-locking to an Rx-side slave laser using injection locking [13] or an optical phase-locked loop (OPLL) [14]. Other approaches involve tapping part of the received signal as carrier reference for an OPLL [15]. In addition, another OPLL has been used to maintain the phase locking. In either solution, the power in a co-propagating pump carrier or the tapped portion of the signal will negatively impact the black-box NF of the PSA.

Here we demonstrate a method that eliminates the need for co-propagating a pump or tapping the received signal before amplification. It uses phase dithering and a field-programmable gate array (FPGA) to control the phase and frequency of an Rx-side pump laser to directly maximise the PSA gain. Our OPLL design is closely related to that of previous works focused on local oscillator phase locking for homodyne coherent reception [16–19]. Here, just like in [18] we add an external phase modulator in addition to the frequency control of the Rx-side laser, the difference being that the PSA itself is used to generate the error signal instead of the phase-sensitive process of coherent reception.

While the PSA performs coherent superposition of waves in the spectral domain (via FWM), a 50/50-coupler (50/50) coherently superposes spatially separate waves, see Fig. 1(b). Similarly to Eq. (1), the 50/50 output power $P_2$ (in Fig. 1) can be written as

Because of its similar phase dependence to the PSA and its simplicity, the 50/50 is used in this work to investigate and understand the OPLL performance under different conditions such as received optical power and loop delay. This is done both experimentally and via numerical simulations. The results following this are then used to predict the performance of the PSA under improved conditions, specifically for reduced loop delay and phase noise of the involved lasers.

Note that this work is an extension of [22] which was presented at Optical Fiber Communication Conference (OFC) in 2023 and where this concept was first presented. In this extension we discuss further details on the optical phase locked loop, what aspects affect its performance and what impact it has on the PSA. We also include the 50/50 phase-locking comparison for OPLL performance evaluation and PSA improvement predictions. In addition, bit error rate (BER) curves have been re-measured with improved results. Also, this work focuses on classical optical communication links and hence, is not in the context of quantum mechanics PSAs as in [23].

In section 2, we present the OPLL achieving the phase lock. The operation of the OPLL on the 50/50 is experimentally and numerically investigated in section 3. The experimental setup for phase-sensitive pre-amplification and the relevant results are presented in section 4. In section 5 we evaluate the OPLL performance on the PSA and discuss the potential of the local pump PSA under improved conditions. Further discussion and conclusion are given in section 6.

2. Optical phase locked loop

To achieve phase sensitive gain in a PSA or coherent superposition in a 50/50, the phases and thereby the frequencies of all input waves must be locked to each other. As was mentioned, for the PSA we focus on we can expect a correlated signal and idler wave while the pump, which is produced at the Rx, will exhibit uncorrelated phase noise and frequency drift (and a possible Doppler shift). The task of the OPLL is therefore to compensate any relative frequency drift or phase noise between these three waves, or in the case of the 50/50, two uncorrelated input waves.

Our OPLL, presented in Fig. 2(a), is applied both to the PSA system in Fig. 2(b) and the 50/50 in Fig. 2(c), where optical amplification ensures similar noise performance in both systems. The OPLL achieves frequency control via current control of a local laser (Rx pump for PSA) and phase control using an external electro-optic phase modulator ($\phi$-mod). After these controls are applied the residual phase error $\phi _e$ ($=\delta \phi,\Delta \phi$ from Eq. (1), (2)) between the waves is minimized during successful locking.

 

Fig. 2. a) Schematic of the OPLL operating on optical system in the dashed box. $\Delta \nu _o$: optical bandpass filter with bandwidth $\Delta \nu _o$. b) PSA sub-system. WDM: wavelength division multiplexer. c) 50/50 sub-system.

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Continuous tracking and minimization of the phase error requires an error signal proportional to $\phi _e$ to be generated. This is done by applying a small single-tone, sinusoidal phase pertubation (dither) $\phi _d\cos (2\pi f_d t)$ in the phase modulator with magnitude $\phi _d$ and frequency $f_d$ ($t$ is time). Via the phase dependence in Eqs. (1) and (2), the dither produces an optical error signal proportional to the phase error $\phi _e$ and oscillating at $f_d$, contained in the power detected in the PD [17]. The electrical signal is then amplified in a low noise amplifier before being fed into a 125 MS/s, 14-bit Red Pitaya field-programmable gate array (FPGA) [24], where sampling, filtering and demodulation (at $f_d$) is carried out. The resulting baseband signal is proportional to $J_1(\phi _d)P_s\sin \phi _e$ which for small phase errors is proportional to the phase error: $\sin \phi _e\approx \phi _e$. Here, $J_m(x)$ is the Bessel function of first kind and order $m$ and $P_s$ is the input signal/idler power for the PSA or power per input for the 50/50 ($P_s=P_\textrm {in,1}=P_\textrm {in,2}$). Finally, $\phi _e$ is minimized as the baseband signal is sent to two separate integrators with equivalent total loop gains $k_\phi$ and $k_f$ for phase control in the electro-optic phase modulator and for frequency control of the local laser, respectively.

Note that due to the implementation of the phase modulator, Whenever its driving voltage limit is reached $2\pi$-phase jumps are carried out to maintain phase-lock. The effect of these ($\sim 1$ $\mu$s duration) phase jumps can be engineered away, e.g., by using an additional phase modulator as in [25] and hence do not pose a fundamental limitation.

While the choice of $\phi _d$, $k_\phi$ and $k_f$ will affect the residual phase error of the OPLL and hence its performance, $f_d$ is typically chosen large enough (here $f_d=31.25$ MHz) for the OPLL to differentiate it from the phase noise that should be compensated.

2.1 Residual phase errors and locking efficiency

Although the OPLL design minimizes $\phi _e$, it will have a residual non-zero value with associated variance $\sigma _{\phi _e}^2$ that will impact the quality of the phase locking and the final PSA performance. Any phase noise beyond the loop bandwidth $B_L$ or system noise within the equivalent noise bandwidth $B_n$, will add contributions $\sigma _\phi ^2$ and $\sigma _\textrm {N}^2$, respectively, to $\sigma _{\phi _e}^2$. Additionally, the phase dither itself constitutes another contribution $\sigma _{\phi _d}^2=\phi _d^2/2$. Other smaller contributors to $\sigma _{\phi _e}^2$ are the impact of finite, discrete frequency resolution $f_\textrm {res}$ in the digital frequency control $\sigma _{f_\textrm {res}}^2$ and the phase error due to frequency drift $\sigma _{\dot {\Delta f}}^2$ (potentially limiting for large Doppler shifts).

The exact variances of all contributions are covered in Supplement 1 upon which the numerical simulations are based. Meanwhile, for the purpose of a qualitative understanding, simpler expressions for the two dominating contributions are obtained assuming Lorentzian phase noise of linewidth $\Delta \nu$, white system noise, small frequency drifts and zero loop delay, for which

From Eq. (3) we note that suppressing $\sigma _\phi ^2$ implies either increasing $k_\phi$ or improving the phase noise characteristics of the involved lasers while $\sigma _\textrm {N}^2$ may be decreased by reducing $k_{\phi }$ or by increasing either the received power $P_s$ or $\phi _d$. For a given $P_s$ and $\Delta \nu$ however, minimizing $\sigma _\textrm {N}^2$ opposes simultaneous minimization of $\sigma _\phi ^2$ or $\sigma _{\phi _d}^2$ via the OPLL parameters $\phi _d$ and $k_\phi$ alone. For a minimal phase error $\sigma _{\phi _e}^2\approx \sigma _{\phi _d}^2+\sigma _\phi ^2+\sigma _N^2$ and best OPLL performance there exists a trade-off between the different contributions and the OPLL parameters $\phi _d$ and $k_\phi$ (and $k_f$ for non-negligible drift) must therefore be optimized.

In a PSA, the impact a residual phase error has is the reduction of phase-sensitive gain and subsequent NF degradation. The important quality metric can therefore be defined in terms of the locking efficiency $\eta (\phi _e)$

3. Locking efficiency of the 50/50-coupler

The simplicity of the 50/50 compared to the PSA makes it a practical testing platform for the OPLL. Thanks to its small size it also allows operating the OPLL at much shorter loop delays than with the 600 m long highly nonlinear fiber PSA. The significant PSA length is further recognized as a potential limitation to OPLL performance since large loop delays in general are problematic in control systems. As such, to investigate the effect of loop delay, the time-average locking efficiency was both experimentally measured and simulated for different excess loop delays $\tau _e$ and the received power per input $P_s$, using the 50/50.

The results are shown in Fig. 3(a)-(d), where, for each investigated combination of $P_s$ and excess loop delay $\tau _e$ the OPLL parameters ($k_\phi$, $k_f$ and $\phi _d$) were optimized to maximize $\langle \eta \rangle$.

 

Fig. 3. $\eta$ and $\sigma _{\phi _e}$ of the 50/50 with standard deviation error bars. a) Measured $\langle \eta \rangle$ vs. $P_s$ for different excess loop delay $\tau _e$ with error signal read-out at output port 1 (destructive) or 2 (constructive). b) Measured and simulated $\langle \eta \rangle$ vs. $\tau _e$ at $P_s=-51.6$ dBm. c) Measured (same as in a)) and simulated $\langle \eta \rangle$ vs. $P_s$ for error read-out at port 2 and different excess loop delay $\tau _e$. d) Corresponding phase error standard deviation $\sigma _{\phi _e}$ from c) but without error bars.

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The measured $\langle \eta \rangle$ was obtained using the setup in Fig. 2(a) and (c) during coherent combining of light from a Thorlabs ULN distributed fiber Bragg grating laser with external feedback cavity (as the local laser) and an NKT fiber laser, both emitting at wavelength $\lambda =1550.5$ nm with sub-kHz linewidths. The locking efficiency was measured over 20 batches as $\langle \eta \rangle =P_2/2P_s$ with $P_2$ being the output port of constructive interference (see Fig. 1(b)), whose power was recorded using a power meter. $P_s$ was swept using a variable optical attenuator (VOA) inserted before the EDFA (with 4 dB NF) and different excess loop delays $\tau _e$ were introduced by fiber delays between the EDFA and the optical filter ($\Delta \nu _o=0.22$ nm). Total loop delays were $\tau _\phi =\tau _{\phi _0}+\tau _e$ and $\tau _f=\tau _{f_0}+\tau _e$, where $\tau _{\phi _0}=1.5$ $\mu$s and $\tau _{f_0}=0.75$ $\mu$s were the delays imposed by the FPGA and radio frequency (RF) system for the phase and frequency control respectively (see Supplement 1). Meanwhile, the frequency resolution $f_\textrm {res}$ was fixed to $930$ Hz.

Simulated $\langle \eta \rangle$ (dashed) were calculated from the recorded OPLL parameter values ($k_\phi$, $k_f$ and $\phi _d$) used in the experiment. For simulated optimum $\langle \eta \rangle$ (dotted) the OPLL parameters were numerically optimized for minimized phase error variance, while simultaneously fulfilling phase margin and SNR conditions to ensure stability (see Supplement 1).

From Fig. 3 it is clear that increasing loop delay has a detrimental effect on the OPLL performance and maximum achievable $\langle \eta \rangle$. In a) the measured $\tau _e=6$ $\mu$s case suffers both a reduction to $\langle \eta \rangle$ vs. the measured $\tau _e=0$ case (from 0.996 to 0.967) as well as a sensitivity reduction, i.e. increased $P_s$ cut-off for when the OPLL loses phase-lock. This cut-off in $P_s$ was also observed to vary depending on the error read-out port, which is expected as the signal-optical noise beating at the PD depends on the signal power, which in the destructive port is negligible. In b) a steady decline in maximum $\langle \eta \rangle$ is observed with increased $\tau _e$ both for experiment and simulated optimum $\langle \eta \rangle$. Increased instability as well as difficulties to achieve and maintain phase-lock was experienced for larger $\tau _e$ in addition to the reduced $\langle \eta \rangle$, to a point where locking was lost occasionally for the largest investigated $\tau _e$ ($\eta =0.759$).

Agreement between numerically calculated $\langle \eta \rangle$ and measurement is good throughout Fig. 3(b-d). Here, the simulated optimum $\langle \eta \rangle$ appears not far above the experimentally obtained efficiencies, indicating a close to optimum performance of the experimental setup. What the simulation appears failing to capture is the increased $P_s$ cut-off due to added excess loop delay. Additional stability conditions are most likely needed for improved comparison.

In general, it is known that large loop gains in combination with large loop delays causes control system instability. The reason for reduced $\langle \eta \rangle$ with increasing delay is therefore explained by the reduced maximum allowable loop gain for maintained stability, which in turn results in a larger $\sigma _\phi ^2$ and worse $\langle \eta \rangle$ than in the $\tau _e=0$ case. Additional simulations showing the individual contributions to $\sigma _{\phi _e}^2$, presented in Supplement 1, support this result.

Lastly, in the case of no excess delay, $\tau _e=0$, measurement results show that the OPLL achieves locking efficiencies $>0.99$, corresponding to NF sensitivity penalties $<0.05$ dB for a PSA. This experimentally verifies the potential of the OPLL, at least for the 50/50 case.

4. Local pump phase sensitively pre-amplified receiver experiment

Having successfully demonstrated coherent combining using the OPLL we now use it here to demonstrate local pump generation in an Rx-side phase sensitively pre-amplified receiver of optical communication data.

In order for a PSA to amplify a signal wave carrying data encoded in its phase $\phi _s$, the phase $\delta \phi$ in Eq. (1) must still be zero. Practically one achieves this by ensuring that the idler phase $\phi _i$ carries the conjugated data signal to cancel out the influence of the data modulation on the PSA operation. Naturally, the idler wave must therefore be created at the Tx where the data is available and subsequently transmitted alongside the signal in the link.

In our experimental setup, depicted in Fig. 4(a), a signal wave carrying a 10 GBaud QPSK data signal is combined in a WDM with an amplified pump wave from the NKT fiber laser. The conjugated idler is then produced by copying the signal using FWM in a highly nonlinear fiber, denoted copier. Note that apart from the data, any phase noise or frequency drift of the signal is also conjugated onto the idler and therefore cancelled out at the PSA. The NKT pump phase noise and frequency drift $\phi _\textrm {NKT}$ is however introduced in the idler as: $\phi _i=2\phi _\textrm {NKT}-\phi _s$.

 

Fig. 4. a) Transmission system with a Tx-side copier stage, a lossy channel, and an Rx-side PSA with either a loop-controlled Rx- or recycled Tx pump. PZT: Piezo-electric transducer, VoA: Variable optical attenuator, MCU: Micro-control unit. b) PSA and PIA output spectra captured by an OSA at $P_s=-56$ dBm.

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After the copier, the pump wave is filtered away using a WDM before signal and idler waves (of equal powers) are sent through a VOA, representing a lossy channel. At the Rx side, the signal and idler are combined with an amplified pump wave (29.4 dBm) from the Thorlabs ULN external fiber-Bragg grating cavity laser with phase $\phi _\textrm {ULN}$. The signal and idler experience gain in the 600 m long highly nonlinear fiber-PSA according to Eq. (1), where now $\delta \phi =2(\phi _\textrm {NKT}-\phi _\textrm {ULN})+\phi _\nu$. The signal is then separated using an optical filter ($\Delta \nu _o=0.22$ nm) before detection in an intradyne coherent receiver and sampling in a real-time oscilloscope for offline processing and BER analysis. Observe that the reception is, in principle, lossless since no signal or idler power needs to be tapped at the input of the amplifier, which would degrade the ”black-box” NF.

The offline digital signal processing involved: resampling, IQ-imbalance and frequency offset compensation, constant modulus, and carrier phase estimation as described in [26]. A 10% tap before coherent reception provided the optical error signal to the OPLL. Polarization controllers throughout the system were used to align the polarizations of all waves at all points.

The received signal/idler power $P_s$ was swept by changing the VOA attenuation and for each $P_s$ the OPLL parameters $k_\phi$, $k_f$ and $\phi _d$ were optimized to maximize the output signal power monitored using an optical spectrum analyzer (OSA). Meanwhile, $f_\textrm {res}$ was halved to $465$ Hz to keep the same effective resolution of 930 Hz (due to $2\phi _p$ in Eq. (1)).

For comparison, PIA measurements were performed where the idler was removed from the PSA input. Similarly, to compare with an ideal PSA, a reference measurement was performed where the correlated pump wave from the Tx was phase-aligned using a PZT (controlled by a micro-control unit (MCU)) and used to pump the PSA.

4.1 Results

Typical PSA output power OSA traces are presented in Fig. 4(b) (here for $P_s=-56$ dBm) where the phase-sensitive operation of the PSA is observed in the added gain compared to the PIA case.

From the received power (measured using a power meter) and the optical signal to noise ratio (OSNR) at the PSA output (measured with the OSA), the gain and noise figure were estimated as; for the local pump-locked PSA ($\textrm {PSA}_\textrm {LP}$) $G_\textrm {LP}=20$ dB and $F_\textrm {LP}=2$ dB, the reference PSA ($\textrm {PSA}_\textrm {REF}$) $G_\textrm {REF}=20.5$ dB and $F_\textrm {REF}=1.5$ dB and for the PIA $G_\textrm {PIA}=14.5$ dB and $F_\textrm {PIA}=4.5$ dB. Note that these NFs are w.r.t the total received power $P_r$, which is always (in this work) $P_r=2P_s$ for the PSA and $P_r=P_s$ for the PIA. Again the less than $3$ dB PSA NF and performance improvement over the PIA indicate phase-sensitive operation of the PSA and successful phase locking of the Rx pump laser.

Presented in Fig. 5 are the BER curves for the $\textrm {PSA}_\textrm {LP}$, $\textrm {PSA}_\textrm {REF}$, and PIA cases along with an EDFA pre-amplified receiver with $F_\textrm {EDFA}=4$ dB as well as theoretical PSA and PIA BER-curves. BERs are plotted vs. $P_r$, in consistence with [22]. Experimental curves are averaged from the 10 best (out of 30) measurement batches of 500 k symbols at each $P_r$. No $2\pi$-phase jumps occurred during these batches.

 

Fig. 5. a) Theoretical and measured BER curves, including standard deviation error bars, vs. total received power $P_r$ (including idler, i.e. $P_r=2P_s$, in case of PSA). Theory PSA and PIA NFs are 0 dB and 3 dB respectively. The sensitivity penalty of $\textrm {PSA}_\textrm {LP}$ w.r.t. $\textrm {PSA}_\textrm {REF}$ and PIA is emphasized in the sub-figure to the right.

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Here we note a close to 3 (2.95)dB sensitivity improvement for the $\textrm {PSA}_\textrm {REF}$ vs. the PIA w.r.t. $P_r$. The power difference of the reference case and the PIA case w.r.t. the two corresponding theoretical curves are both 2.2 dB for BERs below 0.03. The sensitivity penalty at different BER for the $\textrm {PSA}_\textrm {LP}$ can be read from the sub-figure in the right of Fig. 5 w.r.t. $\textrm {PSA}_\textrm {REF}$ in yellow and the PIA when shifted 3 dB to the left, in green. For $P_r>-51$ dBm the $\textrm {PSA}_\textrm {LP}$ sensitivity penalty is $\approx 0.7$ dB vs. $\textrm {PSA}_\textrm {REF}$ and the 3dB-shifted PIA. For $-54 \textrm { dBm}\leq P_r\leq -51$ dBm the corresponding sensitivity penalty is instead $\approx 0.3$ dB. The 0.05 dB power difference between the two curves of the sub-figure of Fig. 5 implies a locking efficiency of $\approx 0.99$ for $\textrm {PSA}_\textrm {REF}$ which incorporates both polarization misalignment and the residual phase error of the slow MCU-based PLL. The $\textrm {PSA}_\textrm {LP}$ sensitivity degradation vs. $\textrm {PSA}_\textrm {REF}$ is therefore solely due to imperfect phase locking of the Rx pump laser whereas penalties vs. the 3dB-shifted PIA also accounts for polarization misalignment.

At higher BERs the phase-estimator of the post-processing no longer converges and outputs erroneous values. Saturation issues within the OPLL is likely the cause for the higher $\textrm {PSA}_\textrm {LP}$ sensitivity penalty at $P_r>-51$ dBm, hence the BER at $-54 \textrm { dBm}\leq P_r\leq -51$ dBm ($-57 \textrm { dBm}\leq P_s\leq -54$ dBm) provides a more relevant result of the OPLL performance. In the next section, this 0.3 dB penalty is further discussed and via simulations observed to improve when either the significant loop delay (600m PSA) or the laser phase noise is reduced.

5. PSA locking efficiency

For the investigated PSA, phase-locking is achieved when the Tx and Rx pump phases coincide $\delta \phi =2(\phi _\textrm {NKT}-\phi _\textrm {ULN})=0$ (neglecting constant phases), hence the coherence of these lasers (the NKT and ULN lasers) govern the phase $\delta \phi$, just as they did $\Delta \phi$ in the coherent combining investigation. The phase noise PSDs of the two lasers, characterized using delayed (80 km) self-heterodyne interferometry [27], are shown in Fig. 6(a) alongside an asymptotic Lorentzian PSD with $\Delta \nu =60$ Hz. A typical relative frequency drift of these lasers is also presented in Fig. 6(b) with corresponding statistics in Fig. 6(c) which indicate a maximum drift of about 1.5 MHz/s.

 

Fig. 6. a) Thorlabs ULN (Rx pump), NKT (Tx pump) and asymptotic Lorentzian phase noise PSDs. b) Typical relative frequency drift between the ULN and NKT lasers. c) Probability density of the drift $\dot {\Delta f}$ in b).

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However, due to the factor of 2 in $\delta \phi =2(\phi _\textrm {NKT}-\phi _\textrm {ULN})$ it follows that the phase noise PSDs of the lasers are experienced 4 times stronger by the PSA. Therefore, $\sigma _\phi ^2$ (as well as $\sigma _{\dot {\Delta f}}^2$) is 4 times larger for the PSA case than for the 50/50. Any phase variation on the PSA input is doubled compared to the 50/50, making phase-locking more challenging. This problem is further exacerbated with increasing loop delay as the need for larger loop gains in conjunction with significant delay renders the OPLL more unstable. Meanwhile, the doubling of $\phi _d$ and $f_\textrm {res}$ can simply be canceled by halving these OPLL parameters.

For the PSA, comparison between experimentally estimated $\langle \eta \rangle$ and simulated optimum $\langle \eta \rangle$ are shown in Fig. 7(a) vs. $P_s$. The locking efficiency $\langle \eta \rangle$ was extracted from the BER curves of Fig. 5, using the PIA results to estimate $E_b/N_0$ for $\phi _e=0$. The locking efficiency was also estimated from the OSA gain measurements using Eq. (1). The BER-estimated $\langle \eta \rangle$ shows the short term efficiency during the 10 best (out of 30) measured symbol batches (of $\sim 1\mu$s duration) whereas the OSA-estimated $\langle \eta \rangle$ was averaged over the OSA sweep time (of $\sim 1$s duration).

 

Fig. 7. a) Estimated $\textrm {PSA}_\textrm {LP}$ $\langle \eta \rangle$ vs. $P_s$ from both BER and OSA gain measurements with standard deviation error bars as well as the simulated optimum $\langle \eta \rangle$ for both the PSA delay $\tau _e=\tau _\textrm {PSA}\approx 3.5$ $\mu$s and $\tau _e=0$. b) Simulated optimum $\langle \eta \rangle$ at $\tau _e=\tau _\textrm {PSA}$ for different phase noise PSD powers $S_\phi$ where the $1S_\phi$ curve is the same as the orange dotted line in a). Dashed curves are without the imposed SNR stability condition.

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As expected, the BER-estimated $\langle \eta \rangle$ follows the same trend as the sensitivity penalty of the subfigure in Fig. 5 and within the marked region ($-57\textrm { dBm}\leq P_s\leq -54$ dBm) $\langle \eta \rangle$ is close to the simulated optimum. This indicates that the PSA experiment performs close to optimal for the given conditions. The OSA-estimated $\langle \eta \rangle$ presents a better estimate of the time-average locking efficiency and is, as expected, slightly lower than that of the BER. It also provides a broader $P_s$ measurement range than the BERs (limited to $P_s<-58$ dBm) and indicates a $P_s$ cut-off of the PSA at $-64$ dBm. The simulated case of $\tau _e=0$ further suggests that a $0.99$ locking efficiency is achievable for a significantly shorter PSA. As sensitivity was observed to degrade with $\tau _e$ in Fig. 3(a) we can expect also an improved sensitivity below $P_s=-64$ dBm in the $\tau _e=0$ case.

In case, when the PSA for performance reasons cannot be made shorter, it is of interest to know the requirement on the lasers to improve $\sigma _\phi ^2$ and thereby the locking performance. A simulation was performed and is presented in Fig. 7(b), showing simulated optimum $\langle \eta \rangle$-curves vs. $P_s$ for different phase noise PSD strengths. Here, $1S_\phi$ is the combined NKT and ULN phase noise PSD and different magnitudes of $S_\phi$ is equivalent to different magnitudes of combined linewidth $\Delta \nu$. From this result, we can predict that a combined phase noise PSD of 8 times lower power ($\Delta \nu$ on the order of 10 Hz) is required to achieve $\langle \eta \rangle \approx 0.99$ given a PSA delay $\tau _e\approx 3.5$ $\mu$s.

In case a significantly shorter PSA is realizable, for instance by implementing it on a photonic integrated circuit [28], the resulting $\tau _e=0$ would alleviate the laser requirements for high $\langle \eta \rangle$. With $\tau _e=0$ the relaxed laser requirements would instead be given by the simulation in Fig. 8(a).

 

Fig. 8. a) Simulated optimum $\langle \eta \rangle$ at $\tau _e=0$ for different phase noise PSD powers. Dashed curves are without the imposed SNR stability condition. b) Analytical optimum $\langle \eta \rangle$ vs. $P_s$ for zero loop delay and negligible drift and frequency resolution. Dash-dot lines are the PSA and solid lines are the 50/50, at different Lorentzian linewidths alongside the laser phase noise PSD case $1S_\phi$ of the lasers used here ($\Delta \nu \approx 350$ Hz).

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Here, the characteristics of the lasers used in this work ($\Delta \nu \sim 100$ Hz) are predicted to achieve $\langle \eta \rangle \geq 0.99$. Intuitively, minimizing the PSA length improves performance and the simulated 0.99 efficiency indicates the potential of the local pump OPLL approach for short PSAs. Yet, worse-performing lasers are not recommended to maintain high locking efficiency, at least for the base delays $\tau _{\phi _0}$ and $\tau _{f_0}$ imposed by the Red Pitaya FPGA and RF low-noise amplifiers used here.

In case the complete loop delay, frequency drift and resolution are negligible and the laser phase noise is purely Lorentzian, the optimum OPLL parameters are found analytically when $\sigma _{\phi _d}^2=\sigma _\phi ^2=\sigma _N^2$ (assuming $\phi _d$ is small) and can be used to calculate the minimum phase error variance for both 50/50 and PSA, as (see Supplement 1)

The corresponding locking efficiency is shown in Fig. 8(b). For larger $P_s$ the 50/50 enjoys higher $\eta$ before $\sigma _{\phi _e}^2$ becomes equal for the two cases (at $P_s=F_n h\nu \Delta \nu _o/8$, here =-59.6 dBm) after which the PSA can reach higher $\eta$ for lower $P_s$. In fact, had we added a polarizer between the optical filter and the PD for the 50/50 it would have aligned with the PSA $\eta$ in the low $P_s$ limit due to the halving of the optical noise-noise beating power in the PD. Eq. (5) and Fig. 8(b) serve as the theoretical limits of what is possible to achieve depending on received signal power and laser linewidth for the OPLL.

6. Discussion and conclusion

Using a dither-based local pump locking scheme we have demonstrated phase-sensitive amplification with up to 5.7 dB sensitivity improvement w.r.t. received signal power (or 2.7 dB NF-reduction) versus the investigated phase-insensitive case.

The effect of loop delay, which was observed to be detrimental to the performance by both measured and simulated results of the 50/50 case, was identified as a limiting factor for our 600 m long PSA. Our numerical simulations were verified accurate w.r.t. experimental results and allows us to to predict the local pump PSA performance under improved conditions. For a 600 m PSA it is predicted that the use of lower phase noise lasers can improve the PSA performance, e.g. reaching $<0.05$ dB sensitivity penalty vs. the ideal PSA case when using $\Delta \nu <10$ Hz linewidth lasers. Moreover, in the case of a significantly shorter PSA, for instance, an integrated solution [28], we predict that the same $<0.05$ dB sensitivity penalty can be reached with lasers such as those used in this work ($\Delta \nu \sim 100$ Hz). Finally, theoretical limits in the zero-loop delay case were also provided.

In terms of received signal powers, the OPLL is demonstrated to maintain phase-lock down to -64 dBm which provides a significant power margin to that required by the data communication itself at the investigated symbol rate (10 GBaud QPSK). For higher symbol rates and consequently higher received powers this margin will grow whereas for much lower received powers the OPLL noise bandwidth will limit the OPLL operation.

In contrast to previous PSA demonstrations [5,15], our system is not penalized by an extra Tx pump carrier reference nor any signal power tap before the PSA. This alleviates both the complexity associated with these solutions as well as the overall ”black-box” NF. The NF-penalty of our local pump solution is dictated by the achieved locking efficiency ($F_\eta =1/\eta$) only. The phase dithering, which comprises the power ”loss” in a dither OPLL, is already accounted for in this efficiency and is a parameter of optimization that provides an extra degree of freedom when designing the OPLL. Although the PSA delay provides a more challenging phase-lock than that of regular OPLLs or injection locking this can be alleviated by improved platforms and equipment.

In conclusion, the demonstrated work provides a simplified and efficient solution with great potential for frequency and phase alignment of signal, idler, and pump waves in shorter PSAs. Not needing to transmit a weak pump reference wave further enables other signal and idler-transmitting schemes such as optical frequency combs and makes PSAs more attractive in fiber- and space communication links.