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Limits on Achievable Dimensional and Photon Efficiencies with Intensity-Modulation and Photon-Counting Due to Non-Ideal Photon-Counter Behavior

Bruce E. Moision, Baris I. Erkmen, William H. Farr, Samuel J. Dolinar, Kevin M. Birnbaum

Published Jan 24, 2012• 0 citations
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Abstract

An ideal intensity-modulated photon-counting channel can achieve unbounded photon information efficiencies (PIEs). However, a number of limitations of a physical system limit the practically achievable PIE. In this paper, we discuss several of these limitations and illustrate their impact on the channel. We show that, for the Poisson channel, noise does not strictly bound PIE, although there is an effective limit, as the dimensional information efficiency goes as e[overline] e PIE beyond a threshold PIE. Since the Holevo limit is bounded in the presence of noise, this illustrates that the Poisson approximation is invalid at large PIE for any number of noise modes. We show that a finite transmitter extinction ratio bounds the achievable PIE to a maximum that is logarithmic in the extinction ratio. We show how detector jitter limits the ability to mitigate noise in the PPM signaling framework. We illustrate a method to model detector blocking when the number of detectors is large, and illustrate mitigation of blocking with spatial spreading and altering. Finally, we illustrate the design of a high photon efficiency system using state-of-the-art photo-detectors and taking all these effects into account.

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An ideal intensity-modulated photon-counting channel can achieve unbounded photon information efficiencies (PIEs). However, a number of limitations of a physical system limit the practically achievable PIE. In this paper, we discuss several of these limitations and illustrate their impact on the channel. We show that, for the Poisson channel, noise does not strictly bound PIE, although there is an effective limit, as the dimensional information efficiency goes as e[overline] e PIE beyond a threshold PIE. Since the Holevo limit is bounded in the presence of noise, this illustrates that the Poisson approximation is invalid at large PIE for any number of noise modes. We show that a finite transmitter extinction ratio bounds the achievable PIE to a maximum that is logarithmic in the extinction ratio. We show how detector jitter limits the ability to mitigate noise in the PPM signaling framework. We illustrate a method to model detector blocking when the number of detectors is large, and illustrate mitigation of blocking with spatial spreading and altering. Finally, we illustrate the design of a high photon efficiency system using state-of-the-art photo-detectors and taking all these effects into account.
Deep-space optical communication links operate under severely limited signal power, approaching the photon-starved regime that requires a receiver capable of measuring individual incoming photons. This makes the photon information efficiency (PIE), i.e., the number of bits that can be retrieved from a single received photon, a relevant figure of merit to characterize data rates achievable in deep-space scenarios. We review theoretical PIE limits assuming a scalable modulation format, such as pulse position modulation (PPM), combined with a photon counting direct detection receiver. For unrestricted signal bandwidth, the attainable PIE is effectively limited by the background noise acquired by the propagating optical signal. The actual PIE limit depends on the effectiveness of the noise rejection mechanism implemented at the receiver, which can be improved by the nonlinear optical technique of quantum pulse gating. Further enhancement is possible by resorting to photon number resolved detection, which improves discrimination of PPM pulses against weak background noise. The results are compared with the ultimate quantum mechanical PIE limit implied by the Gordon–Holevo capacity bound, which takes into account general modulation formats as well as any physically permitted measurement techniques.
Coherent states achieve the Holevo capacity of a pure-loss channel when paired with an optimal measurement, but a physical realization of this measurement is as of yet unknown, and it is also likely to be of high complexity. In this paper, we focus on the photon-counting measurement and study the photon and dimensional efficiencies attainable with modulations over classical- and nonclassical-state alphabets. We first review the state-of-the-art coherent on-off-keying (OOK) with a photon-counting measurement, illustrating its asymptotic inefficiency relative to the Holevo limit. We show that a commonly made Poisson approximation in thermal noise leads to unbounded photon information efficiencies, violating the conjectured Holevo limit. We analyze two binary-modulation architectures that improve upon the dimensional versus photon efficiency tradeoff achievable with conventional OOK. We show that at high photon efficiency these architectures achieve an efficiency tradeoff that differs from the best possible tradeoff-determined by the Holevo capacity-by only a constant factor. The first architecture we analyze is a coherent-state transmitter that relies on feedback from the receiver to control the transmitted energy. The second architecture uses a single-photon number-state source.
Coherent states achieve the Holevo capacity of a pure-loss channel when paired with an optimal measurement, but a physical realization of this measurement is unknown, and likely to be of high complexity. In this paper, we focus on the photon-counting measurement and study the photon and dimensional efficiencies attainable with modulations over classical- and nonclassical-state alphabets. We first review state-of-the-art coherent on-off-keying (OOK) and pulse-position modulation (PPM) with a photon-counting measurement, illustrating its asymptotic inefficiency relative to the Holevo limit. Then we analyze two architectures that improve upon the dimensional versus photon efficiency tradeoff achievable with conventional OOK or PPM. We show that at high photon efficiency these architectures achieve an efficiency tradeoff that differs from the best possible tradeoff by only a constant factor. The first architecture is a coherent-state transmitter that relies on feedback from the receiver to control the transmitted energy. The second architecture uses a single-photon number-state source.
It is well known that ideal free-space optical communication at the quantum limit can have unbounded photon information efficiency (PIE), measured in bits per photon. High PIE comes at a price of low dimensional information efficiency (DIE), measured in bits per spatio-temporal-polarization mode. If only temporal modes are used, then DIE translates directly to bandwidth efficiency. In this paper, the DIE vs. PIE tradeoffs for known modulations and receiver structures are compared to the ultimate quantum limit, and analytic approximations are found in the limit of high PIE. This analysis shows that known structures fall short of the maximum attainable DIE by a factor that increases linearly with PIE for high PIE. The capacity of the Dolinar receiver is derived for binary coherent-state modulations and computed for the case of on off keying (OOK). The DIE vs. PIE tradeoff for this case is improved only slightly compared to OOK with photon counting. An adaptive rule is derived for an additive local oscillator that maximizes the mutual information between a receiver and a transmitter that selects from a set of coherent states. For binary phase-shift keying (BPSK), this is shown to be equivalent to the operation of the Dolinar receiver. The Dolinar receiver is extended to make adaptive measurements on a coded sequence of coherent state symbols. Information from previous measurements is used to adjust the a priori probabilities of the next symbols. The adaptive Dolinar receiver does not improve the DIE vs. PIE tradeoff compared to independent transmission and Dolinar reception of each symbol.
Free-space optical communication systems potentially gain many dBs over RF systems. There is no upper limit on the theoretically achievable photon efficiency when the system is quantum-noise-limited: a) Intensity modulations plus photon counting can achieve arbitrarily high photon efficiency, but with sub-optimal spectral efficiency. b) Quantum-ideal number states can achieve the ultimate capacity in the limit of perfect transmissivity. Appropriate error correction codes are needed to communicate reliably near the capacity limits. Poisson-modeled noises, detector losses, and atmospheric effects must all be accounted for: a) Theoretical models are used to analyze performance degradations. b) Mitigation strategies derived from this analysis are applied to minimize these degradations.
Coherent states achieve the Holevo capacity of a pure-loss channel when paired with an optimal measurement, but a physical realization of this measurement scheme is as of yet unknown, and it is also likely to be of high complexity. In this paper, we focus on the photon-counting measurement and study the photon and dimensional efficiencies attainable with modulations over classical- and nonclassical-state alphabets. We analyze two binary modulation architectures that improve upon the dimensional versus photon efficiency tradeoff achievable with the state-of-the-art coherent-state on-off keying modulation. We show that at high photon efficiency these architectures achieve an efficiency tradeoff that differs from the best possible tradeoff--determined by the Holevo capacity--by only a constant factor. The first architecture we analyze is a coherent-state transmitter that relies on feedback from the receiver to control the transmitted energy. The second architecture uses a single-photon number-state source.