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We report the first demonstration of a near quantum-limited optical homodyne PSK receiver combined with powerful forward-error-correction coding, achieving 1.5 photons/bit sensitivity, within 4.5 dB of the Shannon limit. Phase-locking was achieved at 1.55 µm using an analog dither-based optical phase-locked loop with an external phase modulator. Analysis for this configuration with arbitrary loop damping is given showing a performance advantage for the overdamped case.
©2008 Optical Society of America
1. Introduction
Optical homodyne phase-shift-keying (PSK) modulation is known for its potential energy efficiency, rivaling and even exceeding the sensitivity of photon counting receivers (RXs) in bandwidth-limited systems. Historically, the comparison of various communication systems has been made using the number of photons-per-bit (PPB) required to achieve a 10-9 bit-error rate (BER) without coding. With the advent of modern capacity-approaching codes, another useful metric is the number of PPB required to achieve a BER at the forward-error correction (FEC) threshold, where the code can operate error free. It is well known that the theoretical uncoded sensitivity of optical homodyne PSK is 9 PPB at 10-9 BER and is 3.5 dB better than preamplified differential phase shift keying (DPSK). However, when the comparison takes into account the performance achievable with coding, the sensitivity advantage of optical homodyne is far greater. Fig. 1 highlights the theoretical trade between photon-efficiency and spectral efficiency at channel capacity for several modulation formats [1]. The sensitivity of homodyne PSK with a rate-½ soft-decision (SD) code (bandwidth expansion 2) is approximately 0.5 PPB, 6.7 dB more sensitive than preamplified DPSK and approximately 1 dB more sensitive than photon counting with similar bandwidth expansion. This makes homodyne PSK especially attractive for high-rate satellite, deep-space, and fiber-optic links where both photon and spectral efficiency are important. However, the full potential of optical homodyne has not yet been realized in practice with the best reported coded sensitivity of 5 PPB at 2 Mbps and uncoded sensitivity 2.5 dB from theory at 2 Mbps [2], 3.5 dB at 565 Mbps [3] and 6 dB or more at higher data rates [4].
Fig. 1. Channel capacities for homodyne PSK (soft decision), preamplified DPSK (soft decision) and pulsed photon counting. The total excess bandwidth includes both FEC and modulation bandwidth expansion.
These demonstrations have all used narrow linewidth Nd:YAG lasers at 1.06 µm which cannot leverage the mature telecom infrastructure at 1.55 µm. In this paper we present a rate-½ turbo-coded high-sensitivity optical homodyne PSK RX at 1.55 µm that can leverage telecom-type technologies with coded sensitivity of 1.5 PPB. This is only 4.5 dB from the Shannon-limited channel capacity for a rate-½ code and 2 dB better than capacity for preamplified DPSK shown in Fig. 1.
2. Homodyne receiver design
A sensitive homodyne receiver comprises a number of technically challenging components including high quantum-efficiency (balanced) photodetectors that can accommodate large local oscillator (LO) powers, a low-noise wide-band transimpedance amplifier, a post-detection matched filter and most importantly, the optical phase-locked loop (OPLL) needed to align signal and LO frequency and phase.
While several OPLL designs have been demonstrated [3, 5, 6], our system uses a dither OPLL [7–9]. Applying techniques similar to those widely used in locking Mach-Zehnder modulators and interferometers [10–12], a small dither is applied to the LO phase resulting in amplitude modulation on the recovered base-band data signal at the detector output. The dither signal is recovered by an RF detector and then synchronously detected to generate a phase error signal that drives the LO into lock. The dither OPLL has the simplicity of a balanced loop but locks the LO in-phase or 180° out-of-phase with the modulated signal, thus maximizing the base-band output signal. This eliminates the additional loss and complexity associated with synchronization bit, 90°-hybrid or residual carrier-based implementations used in refs. [3, 5, 6].
The phase tracking error of a second-order PLL with a damping factor of ζ=1/√2 (maximally flat frequency response) has been analyzed previously [9, 13, 14]. We present a more general result that can be applied to PLL designs with other damping factors. This has importance for designs where the phase control bandwidth required is greater than the ability to directly tune the laser (e.g. fiber lasers).
Fig. 2. Simplified model of a second-order phase-locked loop with separate high-frequency and low-frequency control paths.
A simplified model of a second-order phase-locked loop with separate paths for low-frequency laser control and high-frequency external modulator control is shown in Fig. 2. The loop bandwidth is the frequency where the high-frequency loop has a gain of 1, given by
where Kd is the phase detector gain constant in volts-per-radian, Km is the external modulator gain constant in radians-per-volt, and ω2 is the high-frequency integrator gain. We define the crossover frequency, ωc, as the point where the open loop gain transitions from a second order response to a first-order response. This occurs when the two paths have equal gain, resulting in
where Ko is the oscillator gain constant in radians-per-second-volts and ω1 is the low-frequency integrator gain.
The closed loop transfer function is given by
This can also be written in the well known form
where ζ is the damping factor and ωn is the natural frequency.
Using Eq. (3), it can be shown that the residual phase noise contributed by laser linewidth noise in a homodyne receiver has variance of
where Δν is the Lorentzian linewidth FWHM (Hz) of one laser and BL=ωL/2π is the 3-dB bandwidth (Hz) of the second-order phase locked loop. This result, for negligible loop delay, is in agreement with [15] expressed in terms of damping factor and natural frequency where 2ζωn=2πBL, and with prior formulations [9, 13, 14] for the underdamped case of ζ=1/√2. The noise bandwidth, Bn, of a 2nd-order loop is related to BL by
Here, fc=ωc/2π is the frequency (Hz) at which the open loop response transitions from a 2-pole rolloff to a 1-pole rolloff. For ζ=1/√2, the transition frequency fc=BL/2. Using this value for fc in Eq. (6) and combining Eqs. (5) and (6) yields prior formulations.
The residual phase noise contributed by shot noise has variance of [13]
Here, α is the OPLL signal-to-noise ratio (SNR), which for the optical homodyne dither PLL is given by
where Np is the number of PPB, Rb is the channel rate in bits/s, and ϕd is the dither magnitude in radians.
The total phase error variance is the sum of the individual variances including the variance due to the phase dither and is given by
Note here that σe2 is reduced for small fc (overdamped loop). If the loop damping is specified, fc/BL is a constant, r=1/(4ζ 2), and the total phase error variance is minimized when
This result is in agreement with previous analysis for ζ=1/√2 [9]. If fc is allowed to be an independent variable, (in our design fc=3 kHz, constrained by the hardware), the optimum loop bandwidth, BL,opt corresponds to the positive real root of the fourth-order equation
The parameters for our experiment were Δν=5 kHz, Rb=311 Mbps, and fc=3 kHz. For Np=9 PPB, BL,opt=448 kHz which is close to the actual BL of 431 kHz used for our experiment.
Fiber lasers at 1550 nm were used for both signal and LO. The LO laser wavelength tuning was piezo controlled, constrained to approximately 10 kHz bandwidth, with several resonant peaks beginning at 14 kHz as shown in Fig. 3.
The full 431 kHz control bandwidth was achieved by using an external phase modulator, as suggested in [8] and implemented in [16–18] and [19], along with low-bandwidth laser piezo control to track the frequency drift of the two lasers. In [16], an external modulator was used in a self-homodyne experiment which did not require low-frequency control. In [17, 18], high- and low-frequency control was implemented in a homodyne system using narrow linewidth Nd:YAG lasers. Here the high-frequency control provided an additional sensitivity benefit of only 0.8 to 1.3 dB over operation with just the low-frequency control alone. In contrast, the broader linewidth fiber lasers in our system could not be phase locked without the use of both high- and low-frequency control. This approach was also used to stabilize IF frequency in a heterodyne experiment using He-Ne lasers and demonstrated an improvement in frequency stability of four to five orders of magnitude [19].
In our system, the low-frequency (LF) control was implemented with a 2nd-order loop. In parallel with the LF loop is a 1st-order high-frequency (HF) loop driving the external phase modulator, as shown in the block diagram in Fig. 4. We chose a crossover frequency at approximately fc=3 kHz so that the gain of the HF loop dominates the combined loop performance above 10 kHz, where the LF loop has resonances, resulting in stable loop performance. The measured loop response is shown in Fig. 5.
Acquisition is accomplished by disabling the 2nd-order LF loop and tuning the LO wavelength near the signal laser. When the difference frequency of the two lasers is less than the HF loop bandwidth, the loop is re-enabled with the combined control loops pulling into a stable lock.
The range of phase modulator control should be at least 3 to 5 times the expected standard deviation, σm, in order to maintain a low probability of saturation. With an OPLL near optimum, the modulator phase variance σm 2 is expected to be about 3 times the variance due to laser phase noise alone which can be shown to be
It is interesting to note that σ2m,pn does not depend on loop bandwidth, and here fc should be as large as possible to offload the phase modulator.
Two photodetectors were integrated in a balanced configuration with a low-noise transimpedance amplifier (TIA) chip. The measured detection efficiency was 72%, resulting in a 1.4-dB photon-loss penalty. The measured thermal noise penalty of the TIA was 0.3–0.6 dB with a LO photocurrent of 1 mA on each detector and 500 MHz signal bandwidth.
Manchester coding was employed to shift the signal spectrum away from the 1/f-noise region of the amplifiers. With its inherent doubling of the bandwidth, and our desire to operate in the 500-MHz band of low-noise performance, we selected a channel rate of 311 Mbps and a ½-rate-coded information rate at 155.5 Mbps (OC-3). The Manchester coding also reduces the high-to-low bandwidth ratio needed in the low-noise electronics and moves the signal spectrum away from the 4-MHz dither frequency, providing greater isolation of the OPLL from signal crosstalk in its sensitive band of approximately 1 MHz around the dither frequency.
Nearly optimum matched filtering of the square signal waveforms was achieved with a sinc filter implemented with an approach similar to Humblet’s matched optical filter [20], but constructed here with an RF tapped-delay line consisting of two 1:8 power dividers and lengths of semi-rigid coax with an incremental delay of T/8 where T=1/622×106 s. For the Manchester decoder we used another 2-branch tapped-delay line with a relative delay of T seconds followed by a 180° RF hybrid to subtract the outputs of the two arms. The filter response in Fig. 6, measured electrically with a Manchester-coded pulse-pattern generator feeding the filter, shows the nearly ideal triangular matched-filter output waveform.
Fig. 6. Matched-filter response measured electrically with a Manchester-coded pulse-pattern generator feeding the filter.
3. Experimental demonstration
Uncoded performance was measured using a BER tester in real time. Manchester coding renders a system insensitive to pattern length [21], so the 27-1 length Manchester-coded PRBS data source (clocked at 622 MHz) that was used provides a good measure of overall performance. The error detector clock was hard wired at 311 MHz, phase locked to the transmitter clock. We then measured performance to be 3.4 dB from theory at 10-9 BER as shown in Fig. 7. At 10-3 BER, the threshold for moderate codes such as Reed-Solomon (255,239), the performance was only 2.9 dB from theory. Of the 2.9 dB, we can attribute 1.4 dB to photodetector efficiency, 0.3–0.6 dB to amplifier noise, 0.4 dB to phase tracking error and 0.5–0.8 dB for all other effects.
We then measured coded performance using a serially-concatenated code consisting of a rate-½ convolutional outer code, an accumulator inner code, and a 15,120-channel-bit interleaver [22, 23]. Using a 3-GHz bandwidth A/D converter, 3 M samples were acquired at 312.5 Mbps and then off-line processed as in [23, 24] in 100-code-block runs (756,000 bits). The raw channel performance at 312.5 Mbps without the turbo decoder is also shown in Fig. 7. We point out the excellent agreement between the real-time BER measurements and the raw off-line processed channel error rate. At input fluxes below approximately 2 channel PPB, occasional cycle slipping in the OPLL was observed, resulting in excessively high raw channel error rates at several data points shown in Fig. 7. This same data was then processed using the rate-½ turbo decoder resulting in error free performance within the limits of the experiment for optical powers greater than -75.3 dBm, corresponding to a coded receiver sensitivity of 1.5 PPB. The error-free range includes the raw data point (A) corrupted by cycle slipping with a BER much higher than the decoder threshold. Although the code was designed for M-ary PPM without regard to the cycle slipping issues of binary PSK, the structure was found to be well suited for the PSK application. This is a result of the accumulator inner code and interleaver. That is, with this code, a cycle slip is thus seen by the outer code as a single bit error, which is readily corrected. Although such codes have been shown in the literature as correcting overall phase ambiguity, we believe this is the first time that such a coding architecture has been shown to be insensitive to cycle slips, perhaps even frequent cycle slips. Cycle slipping is exacerbated when σe2 becomes large either from large laser linewidths or low signal levels. This experiment shows that it may be possible to operate systems incorporating this coding architecture in regimes with higher σe2 than previously thought possible.
The error-free threshold for the coded system was measured to be 4.5 dB from channel capacity for a rate-½ code at 1.5 PPB. Of the 4.5 dB, we can attribute 1.6 dB to coding loss and 2.9 dB to the hardware implementation. Although this demonstration was done without real-time decoding, commercial soft-decision turbo codec chips are available at this channel rate and demonstrations have been reported at 10 Gbps [25], suitable for use in high-speed telecommunication applications.
Fig. 7 Uncoded performance at 311Mbps and coded performance at 156 Mbps. Uncoded autodyne (self-homodyne) is shown for comparison. RT=real-time, OL=off-line.
4. Conclusions
By combining a near quantum-limited homodyne PSK receiver with powerful rate-1/2 FEC we have demonstrated a 7.8 dB improvement over a quantum-limited uncoded homodyne RX at 10-9 BER. We constructed an optical homodyne Manchester-encoded PSK receiver, using 1.55µm fiber lasers and telecom-type components, achieving a sensitivity of 1.5 photons/bit when combined with strong FEC. This is only 4.5 dB from the Shannon-limited channel capacity for a rate-½ coded system and 2 dB better than theoretical capacity for preamplified DPSK. Photon counting receivers have better theoretical performance with large bandwidth expansions, but our results are comparable to the best sensitivities reported of any receiver type, implemented with only a 4× bandwidth expansion, a capability that is unique to coherent RXs.
Our demonstration utilized an analog dither-based phase-locked loop with an external modulator to achieve high loop bandwidth. We have provided an analysis for this loop configuration with arbitrary loop damping showing a performance advantage for the overdamped case.
We have also shown that a serially-concatenated turbo code that includes a simple accumulator inner code and interleaver, and a rate-½ convolutional outer code can operate error free in the presence of cycle slipping. Although the coding and decoding was performed off-line, these functions can readily be built using available electronics at the data rate of our demonstration. Similar real-time coded systems have been demonstrated at 10 Gbps.
Although the results presented here were targeted for near- and deep-space applications where photon sensitivity is paramount, they also have application in the telecom industry where there is currently great interest in preamplified coherent detection for mitigation of chromatic dispersion effects in fiber networks [15, 26].
Acknowledgments
The authors would like to thank J. Roberge, R. Murphy, T. Yarnall and L. Jeromin for helpful discussions and K. Hoover, B. Romkey, J. Carney, R. Magliocco, L. Hill, and S. Robertson for their assistance. This work was sponsored by the National Aeronautics and Space Administration under Air Force Contract #FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the U.S. Government.
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