Quantum-noise-limited distance of coherent transmission using fiber Raman amplifiers

Kyo Inoue

doi:10.1364/OPTCON.488634

1. Introduction

Long-haul transmission is a permanent target in optical fiber communications. There are several phenomena that degrade the quality of the transmitted signal, which has restricted the transmission distance. Studies have been conducted to overcome or compensate for such degradation. Among the limiting factors, quantum noise is intrinsic and primary, which originates from quantum mechanics and cannot be avoided in principle.

Optical amplifiers are widely used to achieve long transmission distances, which iteratively recover the signal power attenuated during fiber transmission. Such repeating amplifiers induce noise accompanied by signal amplification, which in turn degrades the signal quality. Amplifier noise can be classified into two types: excess noise (classical noise) and intrinsic quantum noise. Excess noise is caused by the non-ideal operating conditions of an amplifier, such as imperfect population inversion (as in the case of an erbium-doped fiber amplifier) and pump fluctuation (as in the case of a Raman amplifier). In contrast, intrinsic quantum noise originates from the quantum properties of light and the amplification medium or the uncertainty principle in quantum mechanics [1]. Although excess noise can be reduced by improving the apparatus, the intrinsic quantum noise is unavoidable because of the principle of quantum mechanics. Ultimately, quantum noise determines the transmission distance of systems that use optical repeating amplifiers.

In current optical transmission systems, erbium-doped fiber amplifiers (EDFAs) are widely used, and their noise characteristics have been well studied [2,3]. The transmission performance determined by quantum noise of phase-insensitive amplifiers or EDFAs was also evaluated in comparison with that of phase-sensitive amplifies [4]. Meanwhile, fiber Raman amplifiers have been studied and developed [5–8]. In Raman-amplified systems, fiber transmission lines are utilized as amplification media, and an approximately distributed-amplification scheme is implementable, which can provide better noise performance than a lumped-amplification scheme such as an EDFA-amplified system. Therefore, it has been generally acknowledged that Raman-amplified systems can achieve a longer transmission distance than EDFA-amplified systems. However, theoretical analysis of Raman amplifiers and their noise properties have mainly considered one-span amplification, not multiple amplifications in optically repeating transmission. The advantage of Raman-amplified systems over EDFA-amplified systems in terms of the transmission distance has not been analyzed quantitatively, to the author’s knowledge. Moreover, a classical approach has been conventionally employed to analyze the noise properties of fiber Raman amplifiers, which phenomenologically includes amplified spontaneous emission (ASE) and evaluates the signal-spontaneous beat noise [5,7] or optical signal-to-noise ratio (OSNR) [8].

Upon the above background, this study theoretically investigates the quantum-limited transmission distance of Raman-amplified repeating systems based on full quantum mechanics. Quantum noise alone is considered to clarify the ultimate and intrinsic performance determined by quantum mechanics, where other impairment factors, such as fiber nonlinearities, bandwidth limitations, and some classical noises, are not considered. A single-channel binary phase-shift keying (BPSK) signal is treated as the fundamental signal format in optical coherent transmission systems. We evaluate the evolution of the signal broadening (or the standard deviation) in the amplitude domain (or the constellation map) as the signal light travels through optically repeating systems, using the transfer functions through an amplifier and passive optical elements derived from full quantum mechanics. Subsequently, an analytical formula for the upper bound of the number of repeating amplifiers is derived. Calculations are then performed using the derived formula. The results quantitatively indicate that Raman-amplified systems can achieve considerably longer transmission distances, more than ten times, than EDFA systems.

2. Theoretical treatment

2.1 BPSK transmission

In this study, we consider a one-channel BPSK signal transmitted over fiber transmission lines while repeatedly amplified at equal intervals. It is the fundamental signal format in optical coherent transmission systems, and the study on this signal format can be readily applied to high-order QAM signals just by narrowing the symbol distance. The fiber attenuation is compensated for by an optical amplifier in each span; that is, the amplifier gain equals the one-span loss. Figure 1 illustrates the BPSK signal in the constellation map, where the mean signal points are symmetrically positioned along the in-phase axis. During the transmission, the mean signal points are maintained at the input or output of each span, while the signal broadening is accumulated with the transmission distance owing to quantum noise. A bit error occurs when a signal point exceeds the vertical line in the middle between the two mean signal points. The probability of the bit error can be evaluated by the Q factor defined as Q ≡ (s₁ – s₀)/(σ ₁ + σ₀) = s/σ, where s_1,0 and σ_1,0 denote the mean signal levels and standard deviations for the binary signal in the in-phase axis, respectively, and s ≡ s₁ = –s₀ and σ ≡ σ₁ = σ₀. In quantum mechanics, the light amplitude is indicated in units of the square root of the photon number. Thus, the mean signal level can be expressed as $s = \sqrt {{n_\textrm{s}}}$ where n_s indicates the mean photon number of the signal light.

Fig. 1. Constellation of BPSK signal. n_s: mean photon number, σ: standard deviation of in-phase amplitude.

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In the following subsections, the analytical formulas are derived based on quantum mechanics, which indicate the BPSK signal transmission distance at which the Q-value reaches a value corresponding to a given bit error rate (BER), e.g., 3.1 for a BER of 10⁻³.

2.2 EDFA

Prior to considering Raman amplification systems, this subsection describes EDFA amplification systems for reference, which are standard optically repeating systems for long-haul transmissions. The system configuration assumed in this subsection is depicted in Fig. 2. Repeating EDFAs are equally placed from the transmitter to receiver, and the transmittance between EDFAs is denoted as T which includes the transmittances of a link fiber and optical elements equipped in an amplifier module such as a WDM coupler for pumping and optical isolators.

Fig. 2. Transmission system model. T_X: transmitter, R_X: receiver, T: transmittance of one span, EDFA#k: the kth EDFA, and N: the number of repeating amplifiers.

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We evaluate the evolution of the variance of the in-phase amplitude with the signal light traveling through the above transmission system. The transmission performance of EDFA-amplified systems has been well studied based on the semi-classical treatment that phenomenologically considers amplified spontaneous emission. In contrast, in this study, expressions for the variance evolution derived from full quantum mechanics, including the intrinsic noise of a coherent state, are employed, in order to analyze the transmission distance in the same framework for Raman-amplified systems presented in the next subsection.

First, the relationship between the variances at the input and output of the kth EDFA, σ_k_.in² and σ_k_.out², respectively, is expressed as follows [1]:

(1)$$\begin{aligned} \sigma _{k.{\text{out}}}^2 &= G(\sigma _{k.{\text{in}}}^2 - \frac{1}{4}) + \frac{1}{2}{n_{{\text{sp}}}}(G - 1) + \frac{1}{4} \hfill \\& = G\sigma _{k.{\text{in}}}^2 + \frac{1}{4}(2{n_{{\text{sp}}}} - 1)(G - 1) \hfill \\ \end{aligned}$$

where G denotes the amplifier gain and n_sp indicates the population inversion parameter or the noise factor. This transfer function is derived from the Heisenberg equation with dipole interaction Hamiltonian including the signal annihilation operator and the energy transition operator in a population-inverted material system [1]. The first, second, and third terms in the first row represent amplified excess noise of the incident light, amplified spontaneous emission (ASE), and intrinsic quantum noise of a coherent state (or vacuum noise), respectively. Owing to the full quantum mechanical treatment, the intrinsic quantum noise is included in the above transfer function, unlike conventional classical treatment wherein ASE is phenomenologically superimposed through an amplifier.

The input at the kth EDFA is the (k – 1)th EDFA output propagating through the link fiber and optical elements, whose variance is expressed as [9–11]

(2)$$\sigma _{k.\textrm{in}}^2 = T\sigma _{k - 1.\textrm{out}}^2 + \frac{1}{4}(1 - T). $$

The first term represents the noise propagating from the (k – 1)th EDFA to the kth EDFA, and the second term represents the vacuum fluctuations accompanied with attenuation through the fiber and optical elements. The above transfer function is obtained based on the quantum-mechanical beam-splitting model using the signal and vacuum field operators [9–11]. Substituting Eqs. (2) into (1) yields the following expression:

(3)$$\begin{aligned} \sigma _{k.{\text{out}}}^2 &= G\{ T\sigma _{k - 1.{\text{in}}}^2 + \frac{1}{4}(1 - T)\} + \frac{1}{4}(2{n_{{\text{sp}}}} - 1)(G - 1) \hfill \\&= \sigma _{k - 1.{\text{in}}}^2 + \frac{1}{2}{n_{{\text{sp}}}}(G - 1) \hfill \\& = \frac{k}{2}{n_{{\text{sp}}}}(G - 1) + \sigma _0^2 \hfill \\ \end{aligned}$$

where TG =1 is applied and σ₀² is the variance at the transmitter.

From Eqs. (2) and (3), the variance of the in-phase amplitude at the receiver, σ_R², is expressed as

(4)$$\sigma _\textrm{R}^2 = T\sigma _{N.\textrm{out}}^2 + \frac{1}{4}(1 - T) = \frac{1}{2}N{n_{\textrm{sp}}}(1 - T) + T\sigma _0^2 + \frac{1}{4}(1 - T), $$

where N denotes the number of repeating amplifiers. From this equation, the Q-value at the receiver, Q_R, can be expressed as follows:

(5)$${Q_\textrm{R}} = \sqrt {\frac{{T{n_0}}}{{B\sigma _\textrm{R}^2}}} = \sqrt {\frac{{T{n_0}}}{{B\{ (1 - T)N{n_{\textrm{sp}}}/2 + T\sigma _0^2 + (1 - T)/4\} }}}, $$

where n₀ indicates the mean signal photon number at the transmitter. Signal bandwidth B is multiplied to the variance in Eq. (5) because the variance in Eqs. (1) – (4) are expressed in the unit frequency.

From Eq. (5), the number of repeating amplifiers N, that provides a given Q-factor at the receiver, can be derived as

(6)$$N = \frac{{1/{n_{\textrm{sp}}}}}{{(1 - T)}}\left\{ {\frac{{2T{n_0}}}{{BQ_\textrm{R}^2}} - 2T\sigma_0^2 - \frac{{1 - T}}{2}} \right\}. $$

In the ideal system condition where the transmission performance is determined by quantum noise alone, n_sp = 1 (i.e., perfect population inversion) and σ₀² = 1/4 (i.e., quantum noise of a coherent state) [1]. By substituting these ideal conditions, Eq. (6) can be rewritten as follows:

(7)$$N = \frac{2}{{(1 - T)}} \cdot \frac{{T{n_0}}}{{BQ_\textrm{R}^2}} - \frac{1}{2}(\frac{T}{{1 - T}} + 1). $$

The upper bound of the number of repeating EDFAs can be evaluated using Eq. (7) as Int[N] for the given system conditions. Subsequently, the transmission distance determined by quantum mechanics is estimated by multiplying Int[N + 1] by the one-span length.

2.3 Raman amplification

Following the EDFA system, we analyze Raman amplification systems in this subsection, assuming the system model illustrated in Fig. 3. Fiber transmission lines are utilized as Raman amplification media, onto which pump lights are launched from repeating nodes placed at equal intervals from the transmitter to receiver. Two pumping schemes are assumed: backward and bidirectional. In both cases, the pumping power is adjusted such that the fiber Raman gain compensates for the losses of the transmission line and repeating node. Subsequently, the signal power launched onto the transmission line from each node is held constant. To facilitate a fair comparison with the EDFA system, the last transmission line in front of the receiver is not Raman pumped.

Fig. 3. System model of Raman-amplified transmission. T_X: transmitter, R_X: receiver, and N: the number of repeating nodes.

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The in-out relationship of the in-phase amplitude variance through a Raman-amplified fiber line is expressed as [12]

(8)$$\sigma _{\textrm{out}}^2 = G(\sigma _{\textrm{in}}^2 - \frac{1}{4}) + \frac{1}{2}{n_\textrm{A}} + \frac{1}{4} = G\sigma _{\textrm{in}}^2 + \frac{1}{2}{n_\textrm{A}} + \frac{1}{4}(1 - G), $$

where G and n_A denote the Raman gain and photon number of the amplified spontaneous emission, respectively. This transfer function is derived based on full quantum mechanics using the Heisenberg equation with the Hamiltonian for Raman interactions and the annihilation operators of the signal light and optical phonons, where the intrinsic quantum noise (or vacuum noise entering through the fiber attenuation) is included.

The Raman gain G in Eq. (8) is expressed as follows:

(9)$$G = \exp \left[ {\int\limits_0^L {(g{n_\textrm{p}}(z) - \alpha )dz} } \right], $$

where g represents the Raman gain coefficient, n_p indicates the pump photon number, α symbolizes the fiber attenuation coefficient for the signal light, z indicates the propagation direction, and L denotes the fiber length. The Raman gain coefficient g is dependent on the pump polarization state relative to that of the signal light in a bulk material. However, in a transmission fiber, the polarization states of the pump and signal lights vary differently during the fiber transmission due to the polarization mode dispersion, and the polarization dependency of the Raman gain is reduced by the averaging effect [7]. Besides, polarization-multiplexed pump lights or depolarized-pump light is available, if necessary, to eliminate the polarization dependency. The Raman gain coefficient g in Eq. (9) is the effective one assuming such averaging effects.

The number of spontaneous photons n_A in Eq. (8) can be expressed as follows:

(10)$${n_\textrm{A}} = {n_\textrm{R}}g\int\limits_0^L {{n_\textrm{p}}({z_0})\exp \left[ {\int\limits_{{z_0}}^L {(g{n_\textrm{p}}(z) - \alpha )dz} } \right]d{z_0}}$$

with

(11)$${n_\textrm{R}} = \frac{1}{{1 - \exp [ - h\Delta f/{k_\textrm{B}}{T_0}]}}, $$

where h represents Planck constant, Δf indicates the frequency difference between the pump and signal lights, k_B symbolizes Boltzmann’s constant, and T₀ is the absolute temperature. Parameter n_R corresponds to the population inversion parameter n_sp in an EDFA.

The pump photon number, n_p(z), in Eqs. (9) and (10) is expressed as n_p(z) = n_p0exp[–α_p(L – z)] for backward pumping and n_p(z) = n_p0{exp[–α_p(L – z)] + exp[–α_pz]} for bidirectional pumping, where n_p0 indicates the pump photon number launched onto a fiber line, and α_p symbolizes the fiber attenuation coefficient at the pump light wavelength. Substituting these expressions for the pump photon number, the Raman gain and the spontaneous photon number in backward pumping are expressed as

(12)$${G_\textrm{b}} = \exp \left[ {\int\limits_0^L {(g{n_{\textrm{p0}}}{e^{ - {\alpha_\textrm{p}}(L - z)}} - \alpha )dz} } \right] = {e^{(g{n_{\textrm{p0}}}{L_{\textrm{eff}}} - \alpha L)}}$$

and

(13)$${n_{\textrm{A(b)}}} = {n_\textrm{R}}g{n_{\textrm{p0}}}\int\limits_0^L {dz \cdot {e^{ - ({\alpha _\textrm{p}} + \alpha )(L - z)}}\exp \left[ {g{n_{\textrm{p0}}}\frac{{1 - {e^{ - {\alpha_\textrm{p}}(L - z)}}}}{{{\alpha_\textrm{p}}}}} \right]}, $$

respectively, and those for the bidirectional pumping are

(14)$${G_{\textrm{fb}}} = {e^{(2g{n_{\textrm{p0}}}{L_{\textrm{eff}}} - \alpha L)}}$$

and

(15)$${n_{\textrm{A(fb)}}} = {n_\textrm{R}}g{n_{\textrm{p0}}}\int\limits_0^L {dz \cdot ({e^{ - {\alpha _\textrm{p}}z}} + {e^{ - {\alpha _\textrm{p}}(L - z)}}){e^{ - \alpha (L - z)}}\exp \left[ {\frac{{g{n_{\textrm{p0}}}}}{{{\alpha_\textrm{p}}}}({e^{ - {\alpha_\textrm{p}}z}} + {e^{ - {\alpha_\textrm{p}}L}} + 1 - {e^{ - {\alpha_\textrm{p}}(L - z)}})} \right]}, $$

respectively, where L_eff ≡ {1 – exp(–α_pL)}/α_p indicates the effective length for the pump light.

Using the above expressions for fiber Raman amplification, we analyze the Raman-amplified transmission system illustrated in Fig. 3. The in-phase amplitude variance at the output of the kth node, σ_k², is expressed as follows:

(16)$$\sigma _k^2 = {T_\textrm{n}}\left\{ {G\sigma_{k\textrm{ - 1}}^2 + \frac{1}{2}{n_\textrm{A}} + \frac{1}{4}(1 - G)} \right\} + \frac{1}{4}(1 - {T_\textrm{n}}), $$

where the brace in the first term represents the variance at the input of the kth node, T_n indicates the node transmittance, and the second term represents the vacuum fluctuation superimposed through the node. When the Raman gain is adjusted to maintain the signal output power constant at each node, i.e., T_nG = 1, Eq. (16) can be rewritten as follows:

(17)$$\sigma _k^2 = \sigma _{k\textrm{ - 1}}^2 + \frac{1}{2}{T_\textrm{n}}{n_\textrm{A}} = \sigma _0^2 + \frac{1}{2}k{T_\textrm{n}}{n_\textrm{A}}, $$

where σ₀² is the variance at the transmitter.

Using Eq. (17), the Q-factor at the receiver can be expressed as follows:

(18)$${Q_\textrm{R}} = \sqrt {\frac{{{T_\textrm{f}}{n_\textrm{0}}}}{{B\{ {T_\textrm{f}}\sigma _K^2 + (1 - {T_\textrm{f}})/4\} }}} = \sqrt {\frac{{{T_\textrm{f}}{n_\textrm{0}}}}{{B\{ N{T_\textrm{f}}{T_\textrm{n}}{n_\textrm{A}}/2 + {T_\textrm{f}}\sigma _\textrm{0}^2 + (1 - {T_\textrm{f}})/4\} }}}, $$

where N indicates the number of repeating nodes and T_f denotes the transmittance of the last transmission line in front of the receiver. From Eq. (18), the number of repeating nodes is expressed as follows:

(19)$$N = \frac{1}{{{T_\textrm{n}}{n_\textrm{A}}}}\left\{ {\frac{{2{n_\textrm{0}}}}{{BQ_\textrm{R}^2}} - 2\sigma_\textrm{0}^2 - \frac{1}{2}(\frac{1}{{{T_\textrm{f}}}} - 1)} \right\}. $$

Under the quantum-noise-limited condition, that is, σ₀² = 1/4, this expression can be rewritten as follows:

(20)$$N = \frac{1}{{{T_\textrm{n}}{n_\textrm{A}}}}\left\{ {\frac{{2{n_\textrm{0}}}}{{BQ_\textrm{R}^2}} - \frac{1}{{2{T_\textrm{f}}}}} \right\}. $$

The number of repeating nodes and, subsequently, the transmission distance can be evaluated by Eq. (20) with Eq. (13) or (15).

3. Calculation

Based on the previous section, we calculated the ultimate transmission distance of Raman-amplified BPSK signal, determined by the quantum noise. First, the transmission distance of the EDFA-amplified systems was calculated as a reference. Figure 4 shows the calculation results obtained using Eq. (7) with Q_R = 3.1 that corresponds to a BER of 10⁻³. In the figure, the upper bound of the number of repeating amplifiers is plotted (solid line) as a function of the transmission losses between the repeating EDFAs labeled in the lower horizontal axis, which includes the losses of the transmission fiber and passive optical elements in the amplifier module, such as optical isolators, couplers, and filters. The upper horizontal axis labels the span length between optical repeaters, which corresponds to the transmission loss labeled in the lower horizontal axis with a 0.2 dB/km fiber attenuation and a 3 dB loss of optical elements. The broken line indicates the corresponding total transmission distance calculated as (number of EDFAs + 1) × (span length). The figure indicates that transmission over several tens of thousands kilometers is achievable, depending on the transmission loss between the amplifiers.

Fig. 4. Number of amplifiers N as a function of transmission loss in EDFA repeating systems. The label in the lower horizontal axis denotes transmission loss between amplifiers, and that in the upper horizontal axis denotes the corresponding one-span length, assuming a node loss of 3 dB and a fiber loss of 0.2 dB/km. The transmission distance, i.e., (one-span length) × (N + 1), is also indicated by broken line. Assumed system conditions are: signal bandwidth B = 40 GHz, Q_R = 3.1 corresponding to a bit error rate of 10⁻³, and the photon number at transmitter (= amplifier output), n₀, is set at a value corresponding to an optical power of 0 dBm.

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The Raman-amplified system was then calculated. The results for the backward and bidirectional pumping are shown in Figs. 5 and 6, respectively, where the number of repeating nodes launching the pump light onto fibers was plotted (solid line) as a function of the one-span fiber length that works as a Raman amplifying medium. The BER at the transmission end was assumed to be 10⁻³. The broken line represents the total transmission distance, assuming a node loss of 3 dB, that is T_n = 0.5, in Eq. (20), for a fair comparison with EDFA systems.

Fig. 5. Number of repeating nodes as a function of one-span length in backward pumped Raman amplifying system. Signal and pump transmission losses are 0.20 and 0.25 dB/km, respectively; frequency difference between the signal and pump lights is 13.2 THz (with which the Raman gain is highest [6]); the absolute temperature is 300 K; the signal photon number at transmitter is set at a value corresponding to an optical power of 0 dBm; Raman gain coefficient and the launched pump photon number, gn_p0, are chosen such that the Raman gain compensates for one-span loss; and node transmittance is T_n = –3 dB.

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Fig. 6. Number of repeating nodes as a function of one-span length in bidirectional pumped Raman amplifying system. System conditions assumed are the same as those in Fig. 5.

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The above calculation results suggest that Raman amplified systems can achieve considerably longer transmission distances than those using EDFAs under quantum-noise-limited conditions, e.g., more than ten times, particularly when using the bidirectional pumping scheme. It is owing to the advantage of distributed amplification over lumped amplification, which has been qualitatively known in general but is quantitatively demonstrated in this work based on full quantum mechanical treatment.

It is noteworthy in Fig. 6 that there is an optimum span length that maximizes the transmission distance in the bidirectional pumping scheme, whereas the transmission distance monotonously decreases with the span length in the EDFA systems, as shown in Fig. 4. Moreover, the variation in the transmission distance as a function of the span length is small compared with the other systems. These features could be attributed to the fact that the bidirectionally pumped Raman amplification transmission is closer to the ideal distributed amplified transmission.

4. Summary and Discussion

This study theoretically investigated the ultimate fiber transmission distance of phase-modulated signal determined by quantum noise in Raman-amplified systems. An analytical formula for the upper bound of the number of repeating amplifiers was derived based on full quantum mechanics, considering the signal fluctuations induced through the amplification and attenuation. Calculations based on the analysis were also presented, the results of which quantitatively indicated that Raman-amplified systems could achieve considerably longer transmission distances, more than ten times, than systems using EDFAs. Although BPSK signal was treated in this paper, the result of the present study can be readily applied to high-order QAM signals by narrowing the symbol distance, and the considerable advantage of Raman systems to EDFA systems would be held in high-order QAM signals.

In this work, full quantum mechanical treatment was employed to strictly evaluate the quantum-noise-limited performance. A similar result may be obtained by a classical treatment such that the accumulation of the noise figures (NFs) of an amplifier and passive optical elements including a transmission fiber are consecutively evaluated [8]. In fact, Ref. [12] calculates the NF of Raman amplification in a 100-km fiber with and without considering intrinsic quantum noise (or vacuum noise), and shows a slight NF difference of 0.5 dB, suggesting that intrinsic quantum noise in the amplification process does not cause serious degradation. However, the classical approach phenomenologically treats noises, which is based on quantum mechanics, such that, for example, the NF of a passive optical element results from vacuum noise. This work employed the rigid and logical approach, which is the base of the phenomenological classical one.

This work considered ideal transmission systems, whose performance is determined by quantum noises alone. Finally, we briefly introduce technical issues or difficulties in fiber Raman amplification [8], which are obstacles to achieve the quantum-limited performances shown in this work.

Double Rayleigh scattering is a critical problem. In Raman scattering, spontaneous emission is generated in the backward direction as well as the forward direction, which is then reflected back in the forward direction by Rayleigh scattering. This double back-scattered spontaneous emission overlaps onto the forwardly propagating signal light and degrades the signal light. This noise light can be suppressed somewhat by inserting isolators into the transmission line; however, it is impractical and unsuitable for the backward pumping scheme.

Another practical problem is encountered in pump fluctuations. Owing to the fact that the Raman gain instantly responses to the pump light intensity, pump light fluctuation causes Raman gain fluctuation, which consequently induces signal fluctuation. Such noise resulting from pump light imperfections is unavoidable for practical devices, particularly when high-power light sources are used for Raman amplification.

The above impairments prevent the realization of quantum-limited transmission performance. In fact, whereas there have been several experimental demonstrations of long-haul transmission using distributed fiber Raman amplification [13–15], the advantage to EDFA systems was not clear, maybe because an experimental comparison in terms of the quantum-limited performance was difficult due to other practical factors. Nevertheless, the ultimate transmission distance determined by quantum noise is considerably longer in Raman systems than in EDFA systems, by more than one order of magnitude, as shown in this work. Therefore, even though quantum-limited systems are difficult to implement in practice, Raman systems still have a potential to achieve notably longer transmission distance than EDFA systems. The results presented in this work suggest that Raman-amplified transmission systems are worthy of challenge for long-haul transmission.

Disclosures

The author declares no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

References

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11. K. Inoue, “Quantum-noise-limited BPSK transmission using gain-saturated phase-sensitive amplifiers,” IEICE Trans. Commun. E104.B(10), 1268–1276 (2021). [CrossRef]

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Quantum-noise-limited distance of coherent transmission using fiber Raman amplifiers

Abstract

1. Introduction

2. Theoretical treatment

2.1 BPSK transmission

2.2 EDFA

2.3 Raman amplification

3. Calculation

4. Summary and Discussion

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (20)

Optics Continuum