## Abstract

Optical communication links are usually made with erbium-doped fiber amplifiers, which amplify the signal waves in a phase-insensitive (PI) manner. They can also be made with parametric fiber amplifiers, in which the signal waves interact with idler waves. If information is transmitted using only the signals, parametric amplifiers are PI and their noise figures are comparable to those of erbium amplifiers. However, transmitting correlated information in the signals and idlers, or copying the signals prior to transmission, allows parametric amplifiers to be phase-sensitive (PS), which lowers their noise figures. The information capacities of two-mode PS links exceed those of the corresponding PI links by 2 b/s-Hz.

©2011 Optical Society of America

## 1. Introduction

As network traffic increases, communication systems need to transmit information at higher rates [1]. Not only should such high-capacity systems encode (transmit) information efficiently [2], they should also transport it efficiently. Communication links are sequences of transmission fibers (attenuators), amplifiers (which compensate fiber losses) and detectors. Standard links employ erbium-doped fiber amplifiers [3] or Raman fiber amplifiers [4], which operate on individual signals in phase-insensitive (PI) manners. However, one could also use parametric amplifiers [5, 6], which are based on four-wave mixing (FWM) in fibers. These devices can operate on individual signals or signal–idler (sideband) pairs, in PI or phase-sensitive (PS) manners.

In a previous paper [7], the transmission and detection of information, and the effects on that information of individual attenuators and parametric amplifiers (or frequency convertors), were studied in detail. In this paper, the methods and results of [7] are reviewed briefly, and used to study the information efficiencies of communication links with PI or PS parametric amplifiers. PS amplifiers have lower noise figures (NFs) than PI amplifiers, which (usually) allow PS links to transport information at higher rates than the corresponding PI links.

The analysis of this paper applies to conventional systems with coherent-state (CS) input signals [8], (complex) amplitude-keyed information formats, linear attenuators and amplifiers, and homodyne detectors. For such systems, the signal and idler (sideband) amplitude fluctuations have Gaussian statistics [9, 10]. This fact has two important consequences. First, the sidebands are specified completely by their amplitude means, variances and correlations. General formulas for the variances and correlations produced by concatenated (multiple-mode) parametric processes are known [11–14]. Second, once the variances and correlations produced by specific links have been determined, the associated information capacities follow from the results of classical information theory [15, 16].

This paper is organized as follows. In Sec. 2, the information capacities of one-mode signals and two-mode signal-idler pairs are defined and discussed briefly. In Sec. 3, the FWM processes that enable PI and PS amplification are reviewed briefly. The noise properties and information capacities of two-mode PI links, one-mode PS links and two-mode PS links are determined in Secs. 4, 5 and 6, respectively, as are the optimal input conditions for these links. Two-mode PS links, which require the inputs to be correlated, have the lowest noise figures and the highest capacities. Finally, in Sec. 7 the main results of this paper are summarized. The measurement of information by homodyne detection was discussed in [7].

## 2. Input information

The analysis of this paper is based on the assumption that the input signals are CS
[8]. For a CS with
amplitude mean 〈*a*〉 =
*α*, where 〈 〉 is an expectation value,
the quadrature mean 〈*q*(*ϕ*)〉
=
(*αe*
^{−}
* ^{iϕ}*
+

*α*

^{*}

*e*)/2

^{iϕ}^{1/2}, where

*ϕ*is the the local-oscillator (LO) phase. 〈

*q*(0)〉 and 〈

*q*(

*π*/2)〉 are called the real and imaginary quadratures, respectively. Quantum optics theory shows that the quadrature variance 〈

*δq*

^{2}〉 = 1/2, which does not depend on the LO phase, and the associated quadrature distribution has Gaussian statistics. The quadrature fluctuations are called vacuum fluctuations, because they do not depend on the amplitude mean. The quadrature signal-to-noise ratio (SNR) is the square of the quadrature mean divided by the quadrature variance. For a CS and an optimal LO phase, the quadrature SNR is 4|

*α*|

^{2}.

In the semi-classical model of fluctuations (noise), one replaces the classical
signal quadrature *x* by the semi-classical quadrature
*y* = *x* + *n*,
where *n* is a Gaussian random variable with mean 0 and (vacuum)
variance *σ _{v}* = 1/2. As shown in the Appendix, this ansatz reproduces exactly the
quadrature variance of a CS with quadrature mean

*x*.

Let *X* =
[*x _{i}*]

*be a vector of signal quadratures and*

^{t}*N*= [

*n*]

_{i}*be a vector of noise quadratures with Gaussian statistics. The information content (capacity) of an ensemble (distribution) of signal vectors is maximal if the signal quadratures also have Gaussian statistics [15]. Such distributions are specified completely by their means, variances and correlations. Define the signal covariance matrix*

^{t}*K*= 〈

_{x}*X X*〉, the noise covariance matrix

^{t}*K*= 〈

_{n}*NN*〉 and the signal-plus-noise covariance matrix

^{t}*K*=

_{y}*K*+

_{x}*K*, and let Δ

_{n}*and Δ*

_{n}*be the determinants of the latter matrices. Then the total information capacity [15]*

_{y}For the special case in which there is only one input quadrature, the covariance matrices are scalars and the input capacity

*σ*= 〈

_{x}*x*

^{2}〉 and the noise strength

*σ*=

_{n}*σ*. This capacity depends logarithmically on the ensemble-averaged SNR

_{v}*σ*/

_{x}*σ*. Hence, any process that changes the SNR (by changing the numerator or the denominator) also changes the capacity.

_{v}For two input quadratures (real and imaginary parts of the same complex amplitude, or quadratures from different amplitudes), the covariance matrices

*σ*= 〈

_{ij}*x*〉. The total capacity depends on the individual SNRs and the (normalized) correlation. If the inputs are independent (

_{i}x_{j}*σ*

_{12}= 0), the total capacity is just the sum of the individual capacities. If the total input strength

*σ*

_{11}+

*σ*

_{22}is constant, the total capacity decreases as the correlation increases. Hence, any process that changes the correlation also changes the total capacity.

## 3. Amplification and attenuation

Amplification and attenuation are governed by linear input–output (IO) equations for the operators of the participating modes [11–14]. Quantum optics theory shows that if the input amplitude fluctuations have Gaussian statistics (as they do for CS), so also do the output amplitude fluctuations [9, 10], which are specified completely by their variances and correlations. As shown in the Appendix, the semi-classical model reproduces exactly the output quadrature variances and correlations produced by sequences of parametric processes.

Amplification is made possible by FWM in highly-nonlinear fibers. The modulation
interaction (MI) is a degenerate FWM process, in which two pump photons are
destroyed, and one signal and one idler photon are created
(2*π _{p}* →

*π*+

_{s}*π*, where

_{i}*π*represents a photon with frequency ω

_{j}*) [17]. In the inverse MI, one photon from each of two pumps is destroyed and two signal photons are created (*

_{j}*π*+

_{p}*π*→ 2

_{q}*π*) [18, 19]. Both processes are illustrated in Fig. 1. The former process provides two-sideband-mode PS amplification if the signal and idler inputs are nonzero, and two-mode PI amplification if the idler input is zero. The latter process always provides one-mode PS amplification.

_{s}Phase conjugation (PC) is a non-degenerate FWM process, in which one photon from each
of two pumps is destroyed, and one signal and one idler photon are created
(*π _{p}* +

*π*→

_{q}*π*+

_{s}*π*) [20]. This process is illustrated in Fig. 2. It provides two-mode PS amplification if both inputs are nonzero and two-mode PI amplification is one input is nonzero. For both two-mode PS processes, the input idler can be produced by another signal generator, or by a PI amplifier used as a copier prior to transmission [21, 22].

_{i}One-mode PS amplification is governed by the IO equations [12, 18]

*y*and

*y′*are input and output (signal-plus-noise) quadratures, respectively, and the subscripts

*r*and

*i*denote real (in-phase) and imaginary (out-of-phase) parts, respectively. The transfer coefficients

*μ*and

*ν*(which can be assumed real) satisfy the auxiliary equation

*μ*

^{2}–

*ν*

^{2}= 1, from which it follows that

*μ*–

*ν*= 1/(

*μ*+

*ν*). The real quadrature is stretched, whereas the imaginary quadrature is squeezed.

Two-mode amplification is governed by the real IO equations [5, 12]

*s*and

*i*, because

*i*already denotes imaginary and

*s*will denote stage number.) The imaginary IO equations are similar (

*r*→

*i*and

*ν*→ –

*ν*). The signal (information-carrying) contributions to the input quadratures combine coherently, whereas the noise contributions add incoherently. For reference, the real superposition modes

*y*

_{±}= (

*y*

_{1}

*±*

_{r}*y*

_{2}

*)/2*

_{r}^{1/2}obey the IO equations

*ν*→ −

*ν*).

Attenuation is modeled as two-mode beam splitting [8, 12], in
which signal photons are converted into loss-mode photons
(*π _{s}* →

*π*). This process is illustrated in Fig. 3. It is governed by the real IO equations

_{l}*l*denotes the loss mode, and the transfer coefficients

*τ*and

*ρ*(which also can be assumed real) satisfy the auxiliary equation

*τ*

^{2}+

*ρ*

^{2}= 1. The imaginary equations are similar (

*r*→

*i*). The input loss modes are vacuum states.

Equations (5) and (7) show that amplification does not decrease the information capacity by itself, because for each quadrature, the signal and noise components are dilated by the same amount, so the quadrature SNRs are not changed [7]. However, amplification combined with attenuation has a significant effect on capacity. First, consider attenuation followed by amplification. By combining Eqs. (5) or (6) with the first of Eqs. (8), one obtains the composite IO equations

*v*

_{±}are the loss-mode quadratures (physical or superposition, for one- and two-mode amplification, respectively). Equations (9) show that attenuation reduces the quadrature capacities, by lowering the quadrature SNRs, but subsequent amplification conserves the (reduced) capacities, for the reason stated above. Now consider amplification followed by attenuation, which is governed by the composite IO equations

*μ*∼

*ν*and

*μτ*∼ 1. Then, for the + mode, the output signal is comparable to the input signal and the attenuator noise is comparable to the transmitted noise. Hence, the output SNR is reduced, but only by a factor of order 1. In contrast, for the – mode, the attenuator noise is much larger than the transmitted noise, and might even be stronger than the transmitted signal, both of which are diminished. Hence, the output SNR is decreased significantly: For all practical purposes, the – mode information is lost. This behavior has important consequences for communication links.

## 4. Two-mode phase-insensitive links

Two-mode PI links are sequences of attenuators followed by two-mode PI amplifiers, as illustrated in Fig. 4. They are considered first, because of their similarity to standard links, which are based on erbium-doped or Raman fiber amplifiers and are also PI. For the first stage in a two-mode PI link, the (real) IO equations are

*τ*and

*ρ*are the transfer coefficients of the attenuator, and

*μ*and

*ν*are the transfer coefficients of the amplifier. The input idler and loss-mode quadratures are denoted by ${v}_{2}^{(1)}$ and ${v}_{l}^{(1)}$, respectively, to emphasize that these inputs are vacuum states, which originate within the stage. After every stage except the last, the output idler is discarded.

By iterating Eq. (11), one finds that
the composite IO equation for an *s*-stage PI link is

*r*, respectively. For the output idler, only the last stage matters, so

For a balanced link (*μτ* = 1), the output
strengths

*L*= 1/

*τ*

^{2}is the stage loss. The noise figure (NF) of the link is defined to be the input SNR divided by the output SNR. It is a figure of demerit. Because the input and output signal strengths are equal, the NF is just the output noise strength divided by the input (vacuum) strength. Equation (18) shows that the NF increases linearly with the number of stages and the loss of each stage. Notice that amplifier and attenuator noise contribute equally to the link NF.

According to Eq. (1), the output signal capacity

*s*(

*L*– 1) ≈ 2

*sL*, which is the NF of the link. This result is similar to the corresponding result for links with erbium-doped fiber amplifiers [23]. However, parametric links also produce an output idler, which is a copy of the signal, and which also contains information, so the idler and total output capacities should also be determined.

It is instructive to consider the last stage in detail. At the end of stage
*s* – 1, the signal and noise strengths are ${\sigma}_{x}^{\left(0\right)}$ and [1 + 2(*s*
– 1)(*L* –
1)]*σ _{v}*, respectively. After the last
attenuation process, ${\sigma}_{x}^{\left(s-1/2\right)}\hspace{0.17em}=\hspace{0.17em}T{\sigma}_{x}^{(0)}$ and ${\sigma}_{n}^{\left(s-1/2\right)}\hspace{0.17em}=\hspace{0.17em}T\left[1\hspace{0.17em}+\hspace{0.17em}2\left(s\hspace{0.17em}-\hspace{0.17em}1\right)\left(L\hspace{0.17em}-\hspace{0.17em}1\right)\right]{\sigma}_{v}\hspace{0.17em}+\hspace{0.17em}\left(1\hspace{0.17em}-\hspace{0.17em}T\right){\sigma}_{v}$, where

*T*=

*τ*

^{2}is the transmission. At this point, the signal capacity

*s*) capacity because loss-mode fluctuations were added, but is higher than the output capacity because idler fluctuations were not (yet) added. Now consider the final amplification process. Equations (6) imply that the output covariance matrices

*s*– 1/2 on

*σ*and

_{x}*σ*were omitted for brevity. It is easy to verify that the determinant Δ

_{n}*=*

_{n}*σ*

_{n}*σ*and, hence, that Δ

_{v}*= (*

_{y}*σ*+

_{x}*σ*)

_{n}*σ*. It follows from these results and Eq. (1) that the output idler capacity

_{v}*L*≫ 1), ${\sigma}_{n}^{\left(s\right)}\hspace{0.17em}\gg \hspace{0.17em}{\sigma}_{v}$, so the idler capacity (23) is only slightly lower than the signal capacity (19): The idler is a very good copy of the signal. It also follows from the aforementioned results that the total output capacity

*ν*

^{2}, so the imaginary sideband and total capacities are the same as their real counterparts.

## 5. One-mode phase-sensitive links

Each stage in a one-mode PS link consists of an attenuator followed by a one-mode PS amplifier, as illustrated in Fig. 5. By combining Eqs. (5) and (8), one finds that the composite IO equation for the first stage is

*y*

^{(0)}and

*y*

^{(1)}are the input and output signal quadratures, respectively, and

*v*

^{(1)}is the input loss-mode quadrature. For the in-phase (real) quadrature the dilation factor

*λ*=

*μ*+

*ν*, whereas for the out-of-phase (imaginary) quadrature

*λ*=

*μ*–

*ν*.

By iterating Eq. (25), one finds that
the composite IO equation for an *s*-stage PS link is

*v*

^{(}

^{r}^{)}is the loss-mode (noise) quadrature associated with stage

*r*. Hence, the output signal and noise strengths are

For a balanced link, the in-phase gain compensates loss
[(*μ* +
*ν*)*τ* = 1], so
the real signal is conserved. The output strengths

*s*(

*L*– 1) ≈

*sL*, which is lower than that of the associated PI link by a factor of 2 (3 dB). It follows from these results and Eq. (1) that the real output capacity

In contrast, even for a balanced link the out-of-phase gain does not compensate loss
[(*μ* –
*ν*)*τ* =
*τ*/(*μ* +
*ν*) =
*τ*
^{2}], so the imaginary signal is
attenuated. The output strengths

*L*) comes from the last stage in the link. Because the output signal strength is lower than the input strength by a factor of

*L*

^{2}

*, the NF of the link is (*

^{s}*L*

^{2}

*+*

^{s}*L*)/(

*L*+ 1) ≈

*L*

^{2}

^{s}^{−1}. Hence, the imaginary output capacity

## 6. Two-mode phase-sensitive links

Two-mode PS links are sequences of attenuators followed by two-mode PS amplifiers, as illustrated in Fig. 6. By combining Eqs. (6) and (8), one finds that the (real) IO equations for the first stage are

*y*

_{1}and

*y*

_{2}are the signal and idler quadratures, respectively, and

*v*and

_{k}*v*are the associated loss-mode quadratures. The other symbols were defined previously.

_{l}By iterating Eqs. (35) and (36), one obtains the composite IO equations [14]

*s*stages. The polynomials

*p*and

_{s}*q*are defined by the initial conditions

_{s}*p*

_{1}=

*μ*and

*q*

_{1}=

*ν*, together with the recursion relations

*p*

_{r}_{+1}=

*μp*+

_{r}*νq*and

_{r}*q*

_{r}_{+1}=

*μq*+

_{r}*νp*. By solving these equations, one finds that

_{r}*σ*is the total input strength and the superscript

_{t}*s*was omitted for brevity. (In this section, we do not abbreviate 〈

*x*〉 by

_{i}x_{j}*σ*, to avoid the use of multiple subscripts.) Because most of the noise variables in Eqs. (37) and (38) are independent, the (common) output noise variance and correlation

_{xij}*n*〉.)

_{i}n_{j}The link is balanced if (*μ* +
*ν*)*τ* = 1. This
condition can be rewritten as *G*
_{0} =
*L*, where *G*
_{0} =
(*μ* + *ν*)^{2} is
the in-phase (power) gain and *L* is the (power) loss of each stage.
By making these substitutions in Eqs.
(42) and (43), and doing
the summations, one finds that

*sL*/2) is lower than that of the associated two-mode PI link by a factor of 4 (6 dB) [14,25]. A difference of 5.5 dB was observed in a recent experiment [26]. For the special case in which

*s*= 1, $\u3008{n}_{j}^{2}\u3009/{\sigma}_{v}\hspace{0.17em}=\hspace{0.17em}\left(L\hspace{0.17em}+\hspace{0.17em}1/L\right)/2$ and 〈

*n*

_{1}

*n*

_{2}〉/

*σ*= (

_{v}*L*– 1/

*L*)/2, which tend to 1 and 0, respectively, as

*L*tends to 1.

The individual and total information capacities are determined by the covariance matrices

*α*=

*σ*/2,

_{t}*β*= [1 +

*s*(

*L*– 1)]

*σ*/2 and

_{v}*γ*= [(1 +

*L*

^{2}

^{s}^{−1})/(

*L*+ 1)]

*σ*/2. (Notice that

_{v}*γ*≪

*β*.) First, the (common) sideband capacity

*α*

^{2}) cancel, so the capacity is determined by smaller terms. This fact requires the variance and correlation calculations to be done accurately.] The sideband capacity is lower than the total capacity (as it must be), but is only slightly lower. (The relative difference is of order 1/

*L*

^{2}.) This means that most (almost all) of the information is shared between the sidebands. They are nearly perfect copies of each other [14]. The common capacity

*σ*/

_{t}*σ*is larger than the SNR for the corresponding PI link by a factor of 2 (equal total input powers) or a factor of 4 (equal input sideband powers). These SNR increases are equivalent to capacity increases of 1/2 or 1 bit per quadrature, respectively. The imaginary quadratures transport information in a similar way.

_{v}sLAlthough the physical-mode calculation is straightforward, the superposition-mode
calculation is instructive. Define the sum and difference modes
*y*
_{±} = (*y*
_{1}
± *y*
_{2})/2^{1/2}, respectively. Then, by
combining Eqs. (35) and (36), one obtains the
superposition-mode IO equations

*s*= 1, $\u3008{n}_{+}^{2}\u3009\hspace{0.17em}=\hspace{0.17em}L$ and $\u3008{n}_{-}^{2}\u3009\hspace{0.17em}=\hspace{0.17em}1/L$, both of which tend to 1 as

*L*tends to 1. By using the identities

*n*

_{1}= (

*n*

_{+}+

*n*

_{–})/2

^{1/2}and

*n*

_{2}= (

*n*

_{+}–

*n*

_{–})/2

^{1/2}, one can show that Eqs. (51) and (52) are equivalent to Eqs. (44) and (45).

It can be shown that the capacity is maximized by using uncorrelated superposition-mode inputs [7]. In this case, the covariance matrices

*α*

_{+}=

*σ*

_{+},

*α*

_{–}=

*σ*

_{–}/

*L*

^{2}

*,*

^{s}*β*

_{+}= [1 +

*s*(

*L*– 1)]

*σ*and

_{v}*β*

_{–}= (1 +

*L*

^{2}

^{s}^{−1})

*σ*/(

_{v}*L*+ 1). Hence, the individual capacities

*C*=

_{t}*C*

_{+}+

*C*

_{–}. If

*α*

_{–}= 0 (as assumed previously),

*C*

_{–}= 0 and

*C*= ln(1 +

_{t}*α*

_{+}/

*β*

_{+})

^{1/2}, which is consistent with the physical-mode result. However, the superposition-mode calculation is easier and the superposition-mode picture provides more physical insight.

The superposition-mode picture also sheds light on PI links. Such links are
inefficient, because they divide the input information equally between the sum and
difference modes, and the difference-mode information is promptly lost. If one
replaces *σ*
_{+} by
*σ _{t}*/2 in the sum-mode capacity, one
recovers the PI signal capacity (in the high-loss regime). So another explanation of
the 3-dB difference between the SNRs associated with PI and PS links is that in the
former, half the information is discarded!

The preceding results apply to links with pairs of CS inputs. For links with
individual CS inputs, one can use PI amplifiers before the links to generate the
idlers (copy the signals). Detailed studies of the noise properties of links with
copiers were made in [14, 25]. Just as copiers increase the link
NFs slightly (by amounts of order 1 ≪ *L*), so also do they
decrease the link capacities slightly.

## 7. Summary

In this paper, detailed studies were made of the information capacities of communication links made with one- and two-mode parametric amplifiers. These studies were based on homodyne detection of (complex) amplitude-keyed signals. The input signals were assumed to be coherent states (CS), which have amplitude fluctuations with Gaussian statistics. If such signals (and their associated idlers) propagate through a sequence of (linear) attenuators and amplifiers, their amplitude statistics remain Gaussian. Hence, the output signal–idler pairs are specified completely by their amplitude means, variances and correlations.

The amount of information (capacity) that can be encoded in an input signal depends
logarithmically on the input signal-to-noise (SNR) ratio, which is the square of the
mean quadrature divided by the quadrature variance. For a CS signal, the quadrature
SNR is 4〈*p*〉, where
〈*p*〉 is the number of photons. For two CS inputs,
the capacity depends on how information is distributed between the signal and idler
(sidebands). If the total input power is fixed, the capacity associated with two
correlated (or anti-correlated) inputs equals that associated with one input, and
the capacity associated with two uncorrelated inputs is higher, by a factor of
almost 2. However, the use of uncorrelated inputs does not maximize the output
information, because of the combined effects of attenuation and amplification on the
sidebands.

Parametric amplification is made possible by four-wave mixing (FWM) in fibers. The degenerate FWM process called inverse modulation interaction (MI) always provides one-mode PS amplification. MI and the non-degenerate FWM process called phase conjugation (PC) provide two-mode PI amplification if the input idler is zero, and two-mode PS amplification if the idler is nonzero. (One can generate such an idler using a second transmitter, or by copying the signal prior to transmission.) Parametric amplification differs from erbium-doped and Raman fiber amplification because it can involve one or two (light) modes and can be PI or PS.

The noise figure (NF) of a communication link is defined as the input SNR divided by
the output SNR. The NF of a (balanced) one-mode PS link is approximately
*sL*, where *s* is the number of stages in the
link and *L* is the loss of each stage. Noise is added by the loss
modes of the fibers (attenuators), which have nonzero input fluctuations. The NF of
a two-mode PI link is approximately 2*sL*. Extra noise is added by
the idlers, which also have nonzero input fluctuations. In contrast, the NF of a
two-mode PS link is approximately *sL*/2. Once again, noise is added
by the attenuators and idlers. However, in the amplifiers the sideband amplitudes
add coherently (factor of 4 increase in strength), whereas the sideband fluctuations
add incoherently (factor of 2 increase), so PS amplification reduces the link NF by
a factor of 4 (6 dB) relative to PI amplification. (An improvement of 5.5 dB was
demonstrated experimentally.) Two-mode links also produce output idlers that are
excellent copies of the signals.

Amplification and attenuation have significant effects on the information carried by a signal. Amplification stretches the in-phase quadrature and squeezes the out-of-phase quadrature. Neither process decreases the capacity by itself (because the coherent and incoherent components are dilated by the same amount). However, the squeezed information produced by amplification is swamped by the noise associated with subsequent attenuation. Because only stretched information survives, it is best to launch information in whichever quadrature(s) will be stretched during propagation. For one-mode amplification, this quadrature is the in-phase (real) quadrature, whereas for two-mode amplification, they are the real sum-mode and imaginary difference-mode quadratures (which correspond to correlated and anti-correlated inputs of equal strength, respectively).

For balanced links, the input and output signal strengths are equal, and the information capacity is limited by the noise added to the signal during propagation. For a one-mode PS link, the real quadrature is transmitted optimally (with only attenuation noise), whereas the imaginary quadrature is attenuated and the information it carries is lost. This is a serious deficiency of one-mode PS links (unless the signals are differential phase-shift keyed or the links incorporate a phase-diversity scheme). For a two-mode PI link, both quadratures are degraded by noise from the attenuators and amplifiers. However, both quadratures are transmitted (neither is attenuated). Although the NF of this (standard) link is higher than that of the corresponding one-mode link, its ability to transport both quadratures gives it a higher capacity. For a two-mode PS link, the real sum-mode quadrature and the imaginary difference-mode quadrature are transmitted optimally (with only attenuation noise), whereas the other quadratures are attenuated and the information they carry is lost. The NFs of two-mode PS links are 6-dB lower than those of their PI counterparts (if the signal powers are equal), which allows them to transport 1 extra bit per quadrature (2 extra bits per mode). To put this result in perspective, an on-off keyed system with a bit rate of 100-Gb/s and a channel spacing of 50-GHz has a spectral efficiency of 2 b/s-Hz. Two-mode PS amplifiers are compatible with multiple-stage (repeatered) links and are well suited to unrepeatered links (festoons).

The preceding discussion of information capacity does not account for nonlinear effects in the transmission fibers (four-wave mixing, and self- and cross-phase modulation). These processes increase the (complex) amplitude fluctuations of the signals, which decrease their information capacities concomitantly [27–30]. The capacity formulas derived herein are useful upper bounds, whose dependences on the system parameters are evident.

## Appendix: Comparison of the quantal and semi-classical results

The coherent state (CS) |*α*〉 is defined by the
eigenvalue equation

*a*is a mode operator,

*α*is a complex parameter and | 〉 is a ket vector. The mode operator satisfies the commutation relation (CR) [

*a*,

*a*

^{†}] = 1, where [ , ] is a commutator and † is a hermitian conjugate. It follows from Eq. (55) that the expectation value 〈

*a*〉 =

*α*, for which reason

*α*is called the mode amplitude. The quadrature operator

*θ*is the local-oscillator (LO) phase, and the quadrature-deviation operator

*δq*(

*θ*) =

*q*(

*θ*) – 〈

*q*(

*θ*)〉. By combining Eqs. (55) and (56) with the CR, one finds that the quadrature mean (first-order moment) 〈

*q*(

*θ*)〉 = (

*αe*

^{−}

*+*

^{iθ}*α*

^{*}

*e*)/2

^{iθ}^{1/2}, which depends on both the mode amplitude and LO phase, and the quadrature variance (second-order moment) 〈

*δq*

^{2}〉 = 1/2, which does not depend on the amplitude or phase.

Multiple-mode parametric processes are governed by the input-output (IO) equations

*a*is an input-mode operator,

_{j}*b*is an output-mode operator, and

_{j}*μ*and

_{jk}*ν*are transfer coefficients. The input modes satisfy the CRs [

_{jk}*a*,

_{j}*a*] = 0 and [

_{k}*a*, ${a}_{k}^{\u2020}$] =

_{j}*δ*, where

_{jk}*δ*is the Kronecker delta function. The output modes satisfy similar CRs, which imply that

_{jk}Suppose that the inputs are CS with amplitudes
*α _{j}*. [If some

*α*= 0, those inputs are vacuum states (VS).] Then the output amplitudes (first-order moments) ${\beta}_{j}\hspace{0.17em}=\hspace{0.17em}{\mathbf{\text{\Sigma}}}_{k}\left({\mu}_{jk}{\alpha}_{j}\hspace{0.17em}+\hspace{0.17em}{\nu}_{jk}{\alpha}_{k}^{*}\right)$. There are two standard ways to calculate the higher-order output moments. In the first method, one combines Eqs. (57) and calculates expectation values using the properties of CS [Eq. (55) and its hermitian conjugate] and the CRs. In the second method, one rewrites the operators as

_{j}*v*and

_{j}*w*also satisfy the CRs and Eqs. (57), and calculates expectation values using the properties of VS (

_{j}*v*|0〉 = 0). The second method will be used herein (because it is similar to the semi-classical method, which will be described shortly).

_{j}The output quadrature and quadrature-deviation operators are also defined by
Eq. (56), with
*a* replaced by *b _{j}* and

*w*, respectively. It follows from these definitions and Eqs. (57) that the output quadratures $\u3008{q}_{j}\left({\theta}_{j}\right)\u3009\hspace{0.17em}=\hspace{0.17em}\left({\beta}_{j}{e}^{-i{\theta}_{j}}\hspace{0.17em}+\hspace{0.17em}{\beta}_{j}^{*}{e}^{i{\theta}_{j}}\right)/{2}^{1/2}$ and the output-quadrature correlations

_{j}*i*=

*j*, the right side of Eq. (63) is manifestly real. When

*i*≠

*j*, the right side of Eq. (63) involves summations of ${\mu}_{ik}{\mu}_{jk}^{*}{e}^{i\left({\theta}_{j}\hspace{0.17em}-\hspace{0.17em}{\theta}_{i}\right)}$,

*μ*

_{ik}ν_{jk}e^{−}

^{i}^{(}

*, ${\nu}_{ik}^{*}{\mu}_{jk}^{*}{e}^{i\left({\theta}_{i}+{\theta}_{j}\right)}$and ${\nu}_{ik}^{*}{\nu}_{jk}{e}^{i\left({\theta}_{i}-{\theta}_{j}\right)}$ Equation (58) implies that the sum of the second and third terms is real, whereas Eq. (59) implies that the sum of the first and fourth terms is real. Hence, the quadrature-correlation formula predicts real correlations (as it must do).*

^{θi+θj)}In the semi-classical method, one adds to each complex amplitude
*α _{j}* a complex random variable

*v*. (Although the same notation is used for quantal operators and classical random variables, the meaning should be clear from the context.) The random variables have Gaussian statistics, and the moments 〈

_{j}*v*〉 = 0, 〈

_{j}*v*〉 = 0 and $\u3008{v}_{j}{v}_{k}^{*}\u3009\hspace{0.17em}=\hspace{0.17em}{\delta}_{jk}/2$. For each input, the quadrature mean 〈

_{j}v_{k}*α*+

_{j}*v*〉 =

_{j}*α*and the quadrature variance 〈

_{j}*v*

^{2}

*e*

^{−}

^{i}^{2}

*+ 2|*

^{θ}*v*|

^{2}+ (

*v*

^{*})

^{2}

*e*

^{i}^{2}

*〉/2 = 1/2, which are exactly the results associated with a CS, and the number mean 〈|*

^{θ}*α*+

_{j}*v*|

_{j}^{2}〉 = |

*α*|

_{j}^{2}+ 1/2, which is approximately the result associated with a CS. Hence, one describes the semi-classical method by saying that 1/2 noise photon is added to each mode. In the semi-classical method, the amplitudes and random variables obey the same IO equations as their quantal counterparts [Eqs. (57) with † replaced by *], so the method produces the same quadrature-correlation formula [Eq. (61)]. It follows from Eqs. (57) and the stated input moments that the output moments

*i*≠

*j*). It underestimates the moment $\u3008{w}_{i}{w}_{i}^{\u2020}\u3009$ by 1/2 and overestimates $\u3008{w}_{i}^{\u2020}{w}_{i}\u3009$ by the same amount. However, the only quadrature moments that depend on these quantities are the variances 〈

*δq*(

_{i}*θ*)

_{i}^{2}〉, which depend on their sums and, hence, are also correct. In summary, the semi-classical method, which consists of adding Gaussian amplitude fluctuations with variance 1/2 to each classical amplitude, reproduces exactly the quadrature means and variances of CS inputs, and the output quadrature means, variances and correlations produced by sequences of parametric processes.

## Acknowledgment

We acknowledge a useful discussion with E. Agrell.

## References and links

**1. **R. W. Tkach, “Scaling optical communications for the
next decade and beyond,” Bell Labs Tech.
J. **14** (4),
3–10 (2010). [CrossRef]

**2. **P. J. Winzer, “Modulation and multiplexing in optical
communication systems,” IEEE LEOS
Newsletter **23** (1),
4–10 (2009).

**3. **P. C. Becker, N. A. Olsson, and J. R. Simpson, *Erbium-Doped Fiber Amplifiers*
(Academic Press,
1999).

**4. **M. N. Islam, *Raman Amplifiers for Telecommunications*
(Springer Verlag,
2003).

**5. **J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric
amplifiers and their applications,” IEEE J.
Sel. Top. Quantum Electron. **8**, 506–520
(2002). [CrossRef]

**6. **S. Radic and C. J. McKinstrie, “Optical amplification and signal
processing in highly-nonlinear optical fiber,”
IEICE Trans. Electron. **E88C**, 859–869
(2005). [CrossRef]

**7. **C. J. McKinstrie and N. Alic, “Information efficiencies of parametric
devices,” to appear in IEEE J. Sel. Top.
Quantum Electron.

**8. **R. Loudon, *The Quantum Theory of Light*, 3rd
Ed. (Oxford University Press,
2000).

**9. **J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in
parametric processes II,” Phys.
Rev. **129**, 481–485
(1963). [CrossRef]

**10. **C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by
parametric processes in fibers,” to appear in
IEEE J. Sel. Top. Quantum Electron.

**11. **C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric
amplifiers driven by two pump waves,” Opt.
Express **12**, 5037–5066
(2004). [CrossRef] [PubMed]

**12. **C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric
processes,” Opt. Express **13**, 4986–5012
(2005). [CrossRef] [PubMed]

**13. **M. Vasilyev, “Distributed phase-sensitive
amplification,” Opt. Express **13**, 7563–7571
(2005). [CrossRef] [PubMed]

**14. **C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number
correlations produced by parametric processes,”
Opt. Express. **18**, 19792–19823
(2010). [CrossRef] [PubMed]

**15. **C. E. Shannon, “A mathematical theory of
communication,” Bell Sys. Tech. J. **28**, 379–423 and
623–656
(1948).

**16. **T. M. Cover and J. A. Thomas, *Elements of Information Theory*, 2nd
Ed. (Wiley,
2006).

**17. **J. Hansryd and P. A. Andrekson, “Broad-band CW-pumped fiber optical
parametric amplifier with 49-dB gain and wavelength-conversion
efficiency,” IEEE Photon. Technol.
Lett. **13**, 194–196
(2001). [CrossRef]

**18. **C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a
fiber,” Opt. Express **12**, 4973–4979
(2004). [CrossRef] [PubMed]

**19. **K. Croussore and G. Li, “Phase regeneration of NRZ-DPSK signals
based on symmetric-pump phase-sensitive
amplification,” IEEE Photon. Technol.
Lett. **19**, 864–866
(2007). [CrossRef]

**20. **S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain
synthesis using nondegenerate-pump four-wave
mixing,” IEEE Photon. Technol. Lett. **14**, 1406–1408
(2002). [CrossRef]

**21. **R. Tang, J. Lasri, P. S. Devgan, V. S. Grigoryan, and P. Kumar, “Gain characteristics of a frequency
nondegenerate phase-sensitive fiber-optic parametric amplifier with phase
self-stabilized input,” Opt.
Express **13**, 10483–10493
(2005). [CrossRef] [PubMed]

**22. **J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a
fiber-optic parametric amplifier in phase-sensitive and phase-insensitive
operation,” Opt. Express **18**, 4130–4137
(2010). [CrossRef] [PubMed]

**23. **E. Desurvire, “Fundamental information-density limits
in optically amplified transmission: an entropy
analysis,” Opt. Lett. **25**, 701–703
(2000). [CrossRef]

**24. **R. Loudon, “Theory of noise accumulation in
optical-amplifier chains,” IEEE J. Quantum
Electron. **21**, 766–773
(1985). [CrossRef]

**25. **Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber
transmission links that use non-degenerate cascaded phase-sensitive
amplifiers,” Opt. Express **18**, 15426–15439
(2010). [CrossRef] [PubMed]

**26. **Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Toward ultra-sensitive optical links
enabled by low-noise phase-sensitive amplifiers,” to
appear in Nat. Photon.

**27. **J. Tang, “The Shannon capacity of dispersion-free
nonlinear optical fiber transmission,” J.
Lightwave Technol. **19**, 1104–1109
(2001). [CrossRef]

**28. **K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, and S. K. Turitsyn, “Information capacity of optical fiber
channels with zero average dispersion,”
Phys. Rev. Lett. **91**, 203901 (2003). [CrossRef] [PubMed]

**29. **R. J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information
transport in fiber-optic networks,” Phys.
Rev. Lett. **101**, 163901 (2008). [CrossRef] [PubMed]

**30. **A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon
limit,” J. Lightwave Technol. **28**, 423–433
(2010). [CrossRef]