We analyze the information efficiency of a deep-space optical communication link with background noise, employing the pulse position modulation (PPM) format and a direct-detection receiver based on Geiger-mode photon counting. The efficiency, quantified using Shannon mutual information, is optimized with respect to the PPM order under the constraint of a given average signal power in simple and complete decoding scenarios. We show that the use of complete decoding, which retrieves information from all combinations of detector photocounts occurring within one PPM frame, allows one to achieve information efficiency scaling as the inverse of the square of the distance, i.e. proportional to the received signal power. This represents a qualitative enhancement compared to simple decoding, which treats multiple photocounts within a single PPM frame as erasures and leads to inverse-quartic scaling with the distance. We provide easily computable formulas for the link performance in the limit of diminishing signal power.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Optical domain offers numerous benefits for deep-space communication compared to the radio frequency band . The primary advantage is access to a much wider bandwidth. Furthermore, the use of laser sources substantially reduces loss due to diffraction of the beam propagating through space, allowing for improved targeting of the emitted signal power. Other technical reasons, such as prospectively lesser in size and weight onboard transmitter modules and the absence of regulatory issues inherent to the use of the radio spectrum additionally make optical communication the technology of choice for future space missions. This motivates a careful study of the performance limits of optical communication links in the photon-starved regime typical for deep-space scenarios.
The standard approach to deep-space optical communication relies on the pulse position modulation (PPM) format shown schematically in Fig. 1(a) which encodes information in symbols defined by the position of a light pulse within a frame of otherwise empty time bins [2–4]. High photon efficiency is achieved by direct detection of the PPM symbols with the help of time-resolved photon counting. In the photon-starved regime some pulses may escape detection, resulting in lower than one probability to generate a click in the bin occupied by a pulse. In the absence of background noise this leads to erasures of input PPM symbols which can be efficiently dealt with using standard error correcting codes . Remarkably, it can be shown that with diminishing average signal power the directly-detected PPM format optimized over the number of time bins within a frame attains the capacity of a narrowband bosonic channel  in the leading order of the power parameter [7–10]. This is associated with unboundedly growing photon information efficiency as the signal power goes to zero.
The above picture becomes much more nuanced when background noise is taken into account. The common conviction is that in this case the maximum attainable transmission rate scales asymptotically as the inverse of the fourth power of the distance between the transmitter and the receiver [11,12], corresponding to vanishing photon information efficiency. This is quadratically worse compared to coherent communication at radio frequencies, for which information rate exhibits inverse-square scaling with the distance in the power-limited regime. Such inverse-square scaling can be viewed as a result of photon information efficiency attaining a constant value, equal to 1 nat or 2 nats (1 nat ≈ 1.44 bits) respectively for shot-noise limited heterodyne or homodyne detection as implied by the Shannon-Hartley theorem .
The purpose of this paper is to examine carefully using Shannon theory  the performance limits of an optical communication link based on the PPM format with direct detection in the presence of background noise. We consider a realistic model of Geiger-mode photon counting detectors which provide only a binary click or no-click outcome in an individual time bin. We demonstrate theoretically that under the constraint of a given average optical power such a system can in principle achieve inverse-square scaling with the distance while offering at the same time high photon information efficiency exceeding the Shannon-Hartley limit. Two ingredients necessary to achieve this regime of operation are identified. The first one is soft decoding strategy which retrieves information from all possible photocount patterns, including multiple clicks within one PPM frame. The second ingredient is the ability to implement the PPM format of an arbitrarily high order, with an unboundedly growing number of time bins within one frame. Under the average power constraint this implies unlimited pulse peak power used in the PPM format. Although this requirement may be incompatible with technical limitations of lasers used in onboard transmitters, we point out that the recently presented concept of structured optical receivers [15–17] enables one to achieve the performance equivalent to the PPM format with evenly distributed instantaneous power of the transmitted optical signal.
This paper is organized as follows. In Sec. 2 we review the relevant parameters of a communication link. The information rate is calculated in Sec. 3. The asymptotic analysis of the information rate in the limit of diminishing signal power is analyzed in Sec. 4. These results are used to discuss quantitatively the range dependence of an exemplary PPM link in Sec. 5. Finally, Sec. 6 concludes the paper.
2. System characteristics
The elementary parameters characterizing the transmitter are the optical power Ptx and the bandwidth B, which defines the duration of an individual time bin as 1/B. Consequently, the average emitted photon number per time bin is Ptx/(Bh fc), where h is Planck’s constant and fc is the signal carrier frequency. Propagation losses and non-unit efficiency ηdet of the detector reduce this figure in a linear manner, which yields the average detected signal photon number na per time bin given byEq. (1) make na scale as r−2 with the range when all other parameters of the link are fixed. We will be interested in the photon-starved regime arising for large distances, when na ≪ 1.
The M-ary PPM format uses M equiprobable symbols corresponding to the location of a single light pulse in a frame of M otherwise empty bins shown in Fig. 1(a). In order to satisfy the average power constraint, the mean photon number in the signal pulse needs to be equal to ns = Mna. Without background noise, direct detection identifies unambiguously the input symbol through the timing information, unless the photon counting detector does not click at all over the duration of the PPM frame. According to the standard theory of photodetection  the probability of such an erasure event is exp(−ns). From the information theoretic viewpoint the communication scheme is described by an M -ary erasure channel with the probability of faithful transmission equal to 1 − exp(−ns).
In the presence of background noise, photocounts may occur also in empty time bins. The noise model considered in this work is based on an assumption that stray light and dark counts generate background whose strength is equivalent to nb photons per time bin and that background counts are statistically independent from each other as well as uncorrelated with the incoming signal. Furthermore, we will take a realistic model for photon counting which discriminates only between the presence or absence of clicks in a given time bin, which applies e.g. to avalanche photodiodes operated in the Geiger mode. Thus the detector generates a click in an empty time bin and a bin occupied by a light pulse with respective probabilitiesFig. 1(b). Other noise models, such as single-mode thermal fluctuations , can by analyzed by replacing Eq. (2) with suitable alternative expressions and following steps described below.
3. Information rate
The most elementary decoding strategy for a noisy link is to interpret as erasures all events when clicks have occurred in multiple time bins within one PPM frame. Such simple decoding would either recover the input PPM symbol, although with a certain error probability induced by background counts, or yield an erasure event. A more general soft decoding strategy is to retrieve information also from sequences containing multiple clicks in individual PPM frames. The maximum attainable transmission rate is given by the Shannon mutual information evaluated for output events taken into consideration. The probability of obtaining a sequence of k clicks in specific bins within one PPM frame is given by one of two expressions
In order to take into account general soft decoding strategies, we will evaluate mutual information per time bin I(K) for a scenario when information is retrieved from sequences containing up to K clicks, while other events are interpreted as erasures. The complete expression reads:
In Fig. 2 we present contour plots of PIE as a function of the PPM order M and the detected signal power na for a fixed background noise power nb = 10−3. Decoding restricted at a fixed level exemplified with K = 1, 2, 5 is compared to the complete decoding scenario when K = M. The qualitative difference between these two cases is clearly seen. While for restricted decoding PIE tends to zero with na → 0, complete decoding enables one to attain a non-zero asymptotic value of PIE with an appropriate choice of the PPM order. This advantage of complete decoding is associated with a divergent asymptotic behaviour of the optimal PPM order M* with the vanishing signal power, shown in Fig. 2 with dashed lines.
In order to gain further insights into the performance of the complete decoding scenario, in Fig. 3 we plot the maximum attainable photon information efficiency PIE* and the corresponding optimal pulse optical energy given byFig. 3(a) indicate the asymptotic values of the PIE calculated using the method presented in Sec. 4. This method provides also the asymptotic values of , indicated with arrows in the inset of Fig. 3(b). Consequently, in the asymptotic limit the optimal PPM order scales inversely with the detected optimal power na as seen in Fig. 2(d). The results for complete decoding are in stark contrast with the simple decoding strategy shown for comparison with dashed lines in Fig. 3. In the latter case, both the photon information efficiency and the optimal pulse energy tend to zero as na → 0. On a side note, it seen in Fig. 3(a) that the PPM format becomes photon inefficient for high optical powers. In this regime the PPM order is fixed and the probability pc that the pulse generates a click approaches one.
4. Asymptotic PIE value
In this section we present a simple method to calculate the asymptotic values of the maximum photon information efficiency PIE∗ and the corresponding optimal pulse optical energy for the complete decoding scenario. The presented results are based on the information theoretic analysis of the channel capacity per unit cost . The starting observation is that the M-ary PPM format can be viewed as a constrained version of generalized on-off keying (OOK) with a binary set of elementary symbols, where a light pulse is sent with a probability 1/M and an empty bin with the probability 1 − 1/M. The constraint has the form of a requirement that every sequence of M consecutive time bins contains exactly one pulse. Because of this constraint, the mutual information I(M) for the completely-decoded PPM link is upper-bounded by the mutual information IOOK for generalized OOK. The latter can be written as Eq. (2) and 20]. For the communication link analyzed here the cost is measured in terms of the optical energy and there is available exactly one input symbol whose cost is equal to zero, namely the empty time bin. In this setting the capacity per unit cost is monotone non-increasing in na and its asymptotic value PIEas in the limit na → 0 can be obtained from the following single-parameter maximization problem: Eq. (2) in terms of two parameters: nb, treated as a given constant, and ns being the optimization variable.
The reasoning presented above implies that PIEas defined in Eq. (9) specifies an upper bound on the photon information efficiency for the PPM format,Appendix that the value PIEas is actually attained in the limit na → 0 by the photon information efficiency of an optimized PPM link in the complete decoding scenario. In Fig. 4 we plot PIEas as a function of the background noise strength nb. The figure also shows the optimal maximizing the right hand side of Eq. (9). Let us recall that specifies the detected optical energy of the pulse. The values PIEas and characterize the attainable long-range performance of a completely-decoded PPM link under a constraint of a fixed average detected signal power. This performance will be discussed in more detail in Sec. 5.
For completeness, we will close this section by analyzing the asymptotic limit na → 0 of the simple decoding scenario. Fig. 3(b) indicates that in the case of simple decoding the optimal pulse energy ns tends to zero with the diminishing average power na. This observation motivates expanding the mutual information I(1) into a power series in ns. The leading order term has quadratic dependence on ns,Fig. 3(a) by the slope of the dashed curves. The corresponding optimal detected pulse energy, given approximately by Fig. 3(b).
5. Range dependence
The maximum information rate R∗ of a PPM link characterized by the bandwidth B can be written as:Fig. 3(a). Assuming for simplicity that the optical pulse employed in the PPM format has a rectangular shape filling the entire time bin with duration equal to B−1, the peak power requirement to attain the optimal performance reads: Eq. (1)hfc is the energy of a single photon at the carrier frequency, and is the optimal detected pulse energy shown in Fig. 3(b).
The link range r enters Eqs. (14) and (15) through the parameters ηtot and na, both defined in Eq. (1). As a numerical example, we have taken the transmitter optical power Ptx = 4 W, the link bandwidth B = 2 GHz, the carrier frequency fc = 2 · 105 GHz, and the transmitter and the receiver antenna diameters respectively Dtx = 0.22 m and Drx = 11.8 m. For this set of parameters, the attainable information rate R*, the optimal PPM order M*, and the required peak power are shown in Fig. 5 with solid lines as a function of the link range expressed in astronomical units (AU) for several values of the background noise parameter nb. For short ranges, below approximately 0.2 AU the performance of the link is limited by the available bandwidth. In this regime the information rate can be characterized by the expression for the noise-free model, given by B · M−1 log2 M. The optimal performance is achieved by the ternary PPM format with M = 3, which gives a slightly higher value of mutual information M−1 log2 M ≈ 0.528 bit/bin compared to either binary (M = 2) or quaternary (M = 4) formats for which mutual information is 0.5 bit per time bin.
For ranges beyond several AU, the information rate R∗ in the complete decoding scenario shown in Fig. 5(a) exhibits a favorable dependence with the distance r following r−2 scaling analogous to the scaling of the detected signal power. This behavior stems from the fact that for diminishing signal power the photon information efficiency PIE∗(na, nb) in Eq. (14) approaches the constant value PIEas depending only on the noise strength nb, and the link range r enters the right hand side of Eq. (14) only through na. Achieving this performance requires the implementation of extremely high PPM orders, as seen in Fig. 5(b). The required PPM order is given by and for large distances it scales as r2, as in this regime the optimal detected pulse energy becomes only a function of the noise strength nb. The same scaling is exhibited by the peak power evaluated according to Eq. (15) and shown in Fig. 5(b).
A more complete analysis of a PPM link would need to include the effects of the detector dead time. However, current superconducting nanowire detectors can have recovery time reduced down to several nanoseconds , which corresponds to just few tens of time bins for an exemplary 2 GHz link. For PPM frames substantially longer than this figure and the probability of a background count over the dead time window much less than one the impact of the detector dead time should be minor, as actual light pulses will still have a good chance to produce counts. Another potential problem is the necessity to generate the signal in the form of infrequent strong light pulses to attain the r−2 scaling of the information rate. This may lower the overall electrical-to-optical power conversion efficiency of the transmitter module, which is essential for downlink space communication. This issue can be resolved by the use of recently proposed structured optical receivers [15–17]. The basic idea is to generate the optical signal with evenly distributed instantaneous optical power in the form of carefully designed phase or phase-and-polarization patterns which enable one to concentrate temporally the signal energy after transmission using optical interference. Such schemes with quasi-cw optical signals can achieve the efficiency of the PPM format at the expense of a more complicated construction and operation of the receiver. However, these are secondary considerations for downlink transmission of data which becomes the main bottleneck in deep-space communication.
The results obtained for the complete decoding strategy are juxtaposed in Fig. 5 with the simple decoding scenario depicted with dashed lines. Most importantly, for long ranges the attainable information rate exhibits disadvantageous r−4 scaling with the distance. This behavior can be easily understood by inserting in Eqs. (14) and (15) in lieu of PIE∗ and the asymptotic expressions for PIE(1) and derived respectively in Eqs. (12) and (13). Because PIE(1) is linear in na, the information rate exhibits quadratic scaling with na implying r−4 dependence on the distance. On the other hand, the optimal PPM order tends to a constant value for large distances and so does the peak power. The numerical difference between complete and simple decoding is significant, for example at r = 10 AU and the background noise strength nb = 10−1 complete decoding allows one to increase the information rate by a factor of hundreds.
We have analyzed the range dependence of a noisy optical communication link employing the PPM format and a Geiger-mode photon counting detector, which produces a binary click or no-click outcome in each elementary time bin. Under a fixed average signal power constraint, the attainable system performance, quantified using Shannon information, dramatically depends on the adopted decoding strategy. In the complete decoding scenario, when information is retrieved from all detection events including sequences containing multiple clicks within one PPM frame, it is in principle possible to achieve r−2 scaling of the information rate with the distance, i.e. the rate becomes directly proportional to the detected signal power. However, reaching the optimum requires a careful adjustment of the PPM order depending on the operating point of the system. The optimal PPM order grows as r2 with the covered distance. The resulting demands on the peak-to-average power ratio of the laser light source in the PPM transmitter can be in principle bypassed by resorting to other modulation formats with a scalable order, such as frequency shift keying , or concentrating optical energy in the time domain after transmission with the help of structured optical receivers [15–17].
In order to derive the lower bound on the photon information efficiency PIE∗ = maxM(I(M)/na) in the complete decoding scenario it will be helpful to resort to an alternative form of the mutual information I(M). Let us denote the count sequence within one PPM frame as y1y2 … yM, where yj = 0 denotes no detector click in the jth time bin, while yj = 1 labels a click in that bin. Further, let p(y|0) withEq. (8) as Eq. (7). Eq. (20) implies that for any M. Let us now insert , where is the value maximizing the right hand side of Eq. (9). This yields: Eq. (9), the above inequality implies that PIEas is the asymptotic value of photon information efficiency also for the optimized PPM format.
TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund.
We acknowledge insightful discussions with C. Heese, M. Jachura, and M. Srinivasan.
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