Optical diversity transmission using WDM signal and phase-conjugate lights through multi-core fiber

Masafumi Koga; Mitsuki Moroi; Hidehiko Takara

doi:10.1364/OE.24.009340

1. Introduction

Internet traffic has been growing over the past few decades, and its growth rate has been rising rapidly in these last few years spurred by the explosive spread of smart-phones and/or mobile devices. To satisfy these ever growing demands cost effectively, network operators need to innovate their networks while minimizing capital and operational expenditures. The innovations expected should enhance the capacity and distance of fiber transmission systems and many research engineers have tackled these issues. The ceiling on transmission capacity and bitrate-distance product possible in single-core, single-mode fiber is currently set at around 100 Tb/s [1,2] and 400 Pb/s*km [3], respectively, by the optical power density limit imposed by fiber fuse [4] and fiber nonlinearity.

Space-division multiplexing (SDM) in multi-core fiber (MCF) is one of the recent advanced topics being researched to increase the transmission capacity per fiber [5]. It can enhance the space density of transmission capacity and several reports have already exceeded 1 Peta-bit/s [6,7]. Unfortunately, the SDM technique cannot overcome the issues of the power limit per core and fiber nonlinearity. The fiber nonlinearity is, as is well known, a factor limiting the Q-value in single-mode fiber transmission. Overcoming the nonlinearity issue would allow us to raise the Q-value, resulting in improving the number of multi-levels in the modulation format and transmission distance, and thus the bitrate-distance product.

The most intensively discussed nonlinearity mitigation technique is digital backward propagation with nonlinear perturbation pre-distortion [8–10]. However, it incurs excessive digital processing cost. Another approach is to use a phase-conjugate light. Several reports model cascade transmission with mid-span phase reversal [11–13]. Optical diversity transmission with mutually phase-conjugate light pair (PCP) and maximum-ratio combination (MRC) is another simple approach to countering fiber nonlinearity [14]. This scheme also enables the optical power density limit to be, in effect, doubled.

We have already demonstrated 4.5dB Q-value enhancement via single-carrier phase-conjugate pair (PCP) in MCF; this approach is based on cancelling the spatially correlated noise (enhanced by the parametric process) and the nonlinear phase-shift (triggered by self-phase modulation) [15]. The cancellation mechanism is expected to cover cross-phase modulation (XPM) induced nonlinear phase shift (NLPS) in wavelength-division multiplexed (WDM) signal transmission [16]. If MRC enhances the Q-value by more than 3dB, the bitrate-distance product of two-core diversity transmission would exceed twice that possible with conventional single-mode, single-core fiber. While PCP transmission with polarization-mode multiplexing has been proposed in references 17 and 18, those papers use the different approach of the coupled nonlinear Schrödinger equation and so cannot address the optical power density issue.

In this paper, we propose optical diversity transmission with PCP and MRC. The MRC of PCP cancels the NLPS induced by XPM. The MRC condition and the approximation of NLPS cancellation are theoretically introduced and their impact is demonstrated in a 2-core diversity experiment on 5 channel WDM transmission using 25 Gbit/s-QPSK PCP over 3-spans of 60km MCF. Superior NLPS cancellation is achieved with CD span-by-span compensation rather than its lumped equivalent at the receiver. The proposal offers Q-value enhancement of better than 4dB, which yields superior bitrate-distance product and optical power density limit performance, compared to twice the single-core transmission in MCF.

This paper is structured as follows. In Section 2, the MRC condition for optical diversity transmission is described along with its theory established for microwave transmission and radar systems [19]. In Section 3, optical field behavior governed by the nonlinear Schrödinger equation is approximated by a fiber transfer function, and a mechanism to cancel the NLPS induced by XPM is given for MCF transmission by the fiber transfer function. In Section 4, the cancellation approximation is confirmed by a 5-channel WDM diversity transmission experiment with PCP.

2. Optical diversity transmission and maximum-ratio combining

MRC based on diversity transmission was originally designed for microwave transmission and radar systems [8], but basically it can be applied to the linear transmission region. We have expanded this theory to cover optical fiber transmission that exhibits non-linear phenomena.

A conceptual diagram of the proposal is depicted in Fig. 1. The optical total power to be transmitted, denoted as P^total, is split M ways. Each component, Pⁱⁿ = P^total/M, is modulated by QPSK and/or M-ary QAM code, and launched into a different core of the same multi-core fiber; the number of cores is M_core ( $M_{c o r e} \geq M$ ). We write the optical signal electric field for core #i input as;

e_{i} (0, t) = \sqrt{2 P_{i}^{i n}} {A^{'}}_{i} (0, t) \cos (ω_{0} t) = (1 / 2) A_{i} (0, t) \exp (j ω_{0} t) + c . c .

where, A_i(0,t) denotes the modulated signal complex amplitude at transmission distance z = 0, ω₀ is its carrier angular frequency, and c.c. stands for the complex conjugate.The Fourier transform of complex amplitude A_i(0,t) is A_i(Ω) with the relationship of

A_{i} (0, t) = \frac{1}{2 π} \int_{- \infty}^{\infty} A_{i} (Ω) e^{j Ω t} d Ω

where, Ω is the signal modulation angular frequency. The time-average of square of the electric field amplitude,

\bar{{| A (0, t) |}^{2}}

, is set at core #i input optical average power as

\bar{{| A (0, t) |}^{2}} = P^{i n}

. This split optical signal is transmitted through the core. The electrical field after propagating a distance of z,

e_{i} (z, t)

, can be obtained from its attenuation and propagation phase delay factor

\exp [- (α / 2) z + j β (ω_{0} + Ω) z]

e_{i} (z, t) = e^{j ω_{0} t} \frac{1}{4 π} \int_{- \infty}^{\infty} A_{i} (Ω) e^{- {(α / 2) + j β (ω_{0} + Ω)} z} e^{j Ω t} d Ω + c . c .

where, α is the attenuation constant,

β (ω_{0} + Ω)

the phase parameter at around carrier frequency ω₀. The phase,

β (ω_{0} + Ω) z

, linearly shifts according to the fiber‘s linear transmission regionwe can expand

β (ω_{0} + Ω)

near the carrier frequency into a Taylor series

β_{i} (ω = ω_{0} + Ω) = β_{i} (ω_{0}) + {\frac{\partial β_{i}}{\partial ω} |}_{ω_{0}} Ω + \frac{1}{2} {\frac{\partial^{2} β_{i}}{\partial ω^{2}} |}_{ω_{0}} Ω^{2} + \dots

and obtain

\begin{array}{l} e_{i} (z, t) = (1 / 2) \exp [j (ω_{0} t - β_{0 i} z)] \times \\ [\frac{1}{2 π} \int_{- \infty}^{\infty} A_{i} (Ω) \exp {- (α_{i} / 2) + j (Ω t - \frac{Ω z}{v_{g i}} - \frac{1}{2} \frac{d}{d ω} (\frac{1}{v_{g i}}) Ω^{2} z)} d Ω] + c . c . \end{array}

as the approximation up to the second order terms, where,

β_{0, i} = β_{i} (ω_{0})

,

{d β_{i} / d ω |}_{ω_{0}} = 1 / v_{g i}

(group velocity). Here we denote the linear transfer function of core # i (i = 1, 2, …, M) branch transmission channel as,

H_{i}^{L} (Ω) = \exp {- j Ω (1 / v_{g} + b_{i} Ω) z}

where, b_i is the group velocity dispersion(GVD) defined as

b = (d^{2} β / d ω^{2}) / 2

. The output electrical field of core # i is given by

Fig. 1 Optical diversity transmission and maximum-ratio combining model; Pⁱⁿ: Fiber input power, H_i ^L: channel #i linear transfer function, Z_i: Fourier transform of white Gaussian noise, W_i: Weighting function.

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e_{i} (z, t) = \exp [- (α_{i} / 2) + j (ω_{0} t - β_{0 i} z)] \frac{1}{4 π} \int_{- \infty}^{\infty} A_{i} (Ω) H_{i}^{L} (Ω) \exp {j (Ω t)} d Ω + c . c .

The M divided signals are combined after weighting by w_i(t). A white Gaussian noise z_i(t) such as amplified spontaneous emission noise is added to the optical signal in each channel. Here, we define Fourier spectrum Z_i(Ω) and W_i(Ω) to be the Fourier transform of z_i(t) and w_i(t), respectively.

After combining the M split optical signals with weighting, we obtain;

\begin{array}{l} A^{o u t} (Ω) = \sum_{i = 1}^{M} {W_{i} (Ω) (H_{i}^{L} (Ω) A_{i} (Ω) + Z_{i} (Ω))} \\ = \sum_{i = 1}^{M} {W_{i} (Ω) H_{i}^{L} (Ω) A_{i} (Ω)} + \sum_{i = 1}^{M} {W_{i} (Ω) Z_{i} (Ω)} \\ = (W^{T} H) A + W^{T} Z \end{array}

where,

W^{T} = [W_{1} (Ω), W_{2} (Ω), \dots, W_{M} (Ω)]

,

H^{T} = [H_{1}^{L} (Ω), H_{2}^{L} (Ω), \dots, H_{M}^{L} (Ω)]

,

A^{T} = = [A_{1} (Ω), A_{2} (Ω), \dots, A_{M} (Ω)]

. The output signal power, except noise, becomes;

\begin{array}{l} P^{o u t} = E [(W^{T} H A) {(W^{T} H A)}^{*}] = (W^{T} H) E [A A^{*}] {(W^{T} H)}^{*} \\ = (W^{T} H) {(W^{T} H)}^{*} E [A A^{*}] = (W^{T} H) {(W^{T} H)}^{*} (P^{t o t a l} / M) \end{array}

where, E[ ] denotes ensemble average. The total noise power Z^total is written as follows, assuming linear transmission as

z_{k} (t)

terms are mutually independent;

\begin{array}{l} Z^{t o t a l} = E [(\sum_{i = 1}^{M} W_{i} Z_{i}) {(\sum_{i = 1}^{M} W_{i} Z_{i})}^{*}] = (W^{T} Z) (^{W^{T} Z) *} = W^{*} W N_{i} \\ = \sum_{i = 1}^{M} {| W_{i} |}^{2} N_{i} \end{array}

where, N_i is core #i branch-noise power. Taking the ratio of Eq. (9) to (10), yields SNR Γ as;

Γ = \frac{P^{o u t}}{Z^{t o t a l}} = \frac{(W^{T} H) {(W^{T} H)}^{*} (P^{t o t a l} / M)}{W^{*} W N_{i}} = \frac{{| \sum_{i}^{M} W_{i} H_{i} |}^{2}}{\sum_{i} {| W_{i} |}^{2} N_{i}} (P^{t o t a l} / M)

By applying the Schwarz Inequality for complex-valued numbers, this SNR is maximized to

Γ_{\max} = (P^{t o t a l} / M) \sum_{i}^{M} {| H_{i} |}^{2} / N_{i}

if

W_{i} = a H_{i}^{L *} (Ω) / N_{i}

is satisfied, where a is some arbitrary complex constant. This condition means that the optimum weight for each core branch is inversely proportional to the branch-noise power and proportional to the complex-conjugate of the fiber transfer function. In other words, to achieve optimum SNR, the channel with worst noise figure should be most lightly weighted. Considering the phase term, the GVD,

\exp (- j b_{i} Ω^{2} z)

, should be compensated and the propagation phase delay

\exp (j Ω z / v_{g})

should also be adjusted in some manner.

3. Transfer function approximation

3.1 L-span repeat WDM transmission

We rewrite Eq. (1) for the k-th channel optical electrical field in WDM transmission through core # i by adding subscript k as ;

e_{i, k} (0, t) = (1 / 2) A_{i, k} (0, t) \exp (j ω_{k} t) + c . c .

where, ω _k denotes the k-th channel (ch) optical carrier angular frequency among WDM signals. The WDM signals are, in practice, transmitted through an L-span fiber-transmission system with Erbium-doped fiber amplifier (EDFA) repeaters, as shown in Fig. 2; we assume equal span length of z_r. Here we denote the complex amplitude for the k-th channel optical field at the l-th repeater EDFA output as

A_{i, k}^{l} {(l - 1) z_{r}, t}

and its Fourier transform as

{\tilde{A}}_{i, k}^{l - 1} (Ω)

with therelationship of

A_{i, k}^{l} {(l - 1) z_{r}, t} = \frac{1}{2 π} \int_{- \infty}^{\infty} G^{l} (Ω) {\tilde{A}}_{i, k}^{l - 1} (Ω) e^{j Ω t} d Ω

where,

G^{l} (Ω)

_, the l-th repeater gain, is equal to the fiber propagation loss of

\exp {- (α_{i} / 2) z_{r}}

. Then, at the (l + 1) ^th repeater input after propagating the span length of z_r, the complex amplitude

{\tilde{A}}_{i, k}^{l} (Ω)

is given by;

{\tilde{A}}_{i, k}^{l} (Ω) = {\tilde{A}}_{i, k}^{l - 1} (Ω) H_{i, k}^{L} (Ω)

where,

H_{i, k} (Ω)

corresponds to the fiber transfer function for the span length of z_r. When the GVD is compensated in span-by-span manner (S-by-S) by utilizing dispersion compensation fiber (DCF), Eq. (16) becomes

{\tilde{A}}_{i, k}^{l} (Ω) = {\tilde{A}}_{i, k}^{l - 1} (Ω) H_{i, k}^{L} (Ω) \exp {j b_{i, k} Ω^{2} z_{r}}

Then the optical field after propagating the repeated distance of z_r from l-th EDFA output to the (l + 1) ^th input,

e_{i, k} (l z, t)

is, similar to the relationship of Eq. (2) and (7), given by;

\begin{array}{l} e_{i, k} (l z, t) = \exp [- (α_{i} / 2) z_{r} + j {ω_{0} t - β_{0 i, k} \cdot (l z_{r})}] \times \\ \frac{1}{4 π} \int_{- \infty}^{\infty} {\tilde{A}}_{i, k}^{l} (Ω) H_{i, k}^{L} (Ω) e^{j Ω t} d Ω + c . c . \end{array}

where,

β_{0 i, k}

denotes the propagation constant.

Fig. 2 L-span transmission model for transfer function approximation; DCF; Dispersion compensation fiber, G^l : lth repeater amplifier gain..

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In WDM transmission, XPM is the major factor distorting the waveform via nonlinear phase shift. Here we introduce the nonlinear phase shift term due to XPM as

H_{i, k}^{N L} (Ω) = \exp {- j β_{i, k}^{N L} (Ω) z_{r}}

β_{i, k}^{N L} (Ω) \equiv 2 γ_{i} \sum_{j = 1, j \neq k}^{J} {| A_{i, j} (Ω) |}^{2}

where,

γ_{i}

is the nonlinearity coefficient of core # i. Note that this paper focuses on the XPM induced by simple WDM transmission and evaluates its impact on the worst-case nonlinear phase-shift, thus we use a factor of 2 instead of 16/9 for

γ_{i}

, which is generally used in studies that consider polarization fluctuation. Since

β_{i, k}^{N L} (Ω) z_{r}

varies with time because of its optical power dependency, it represents the time-averaged nonlinear phase-shift. Thus we can approximate, for core #i, the k-th channel fiber transfer function

H_{i, k} (Ω)

as the product of the linear transfer function

H_{i, k}^{L} (Ω)

and the nonlinear transfer function

H_{i, k}^{N L} (Ω)

as follows;

H_{i, k} (Ω) = H_{i, k}^{L} (Ω) H_{i, k}^{N L} (Ω)

Then

H_{i}^{L *} (Ω)

in Eq. (13) is replaced by

H_{i, k}^{*} (Ω) = \exp {j Ω (1 / v_{g i, k} + b_{i, k} Ω) z_{r}} \exp {j β_{i, k}^{N L} (Ω) z_{r}}

Generally speaking, the compensation of

β_{i, k}^{N L} (Ω) z_{r}

is not so easy, and there are several approaches to mitigate nonlinear signal interaction. The most intensively discussed mitigation technique up to now is digital backward propagation with nonlinear perturbation pre-distortion. However, it incurs excessive digital processing overheads. The other way is to use a phase-conjugate light. Several reports have examined modeling cascade transmission with mid-span phase reversal. The other approach, described below, is optical diversity transmission with maximum-ratio combining of the WDM signal and its conjugate lights pair.

The combining of a signal and its phase-conjugate light approximately cancels the nonlinear phase-shift induced by the parametric process automatically without any calculation. This approach well suits optical diversity transmission with MRC. In the next section, we will describe a mechanism to cancel the nonlinear phase-shift induced by XPM, the major issue limiting WDM transmission performance.

3. 2 Nonlinear phase-shift cancellation by WDM phase-conjugate light pair

When we transmit quadrature-phase shift keying (QPSK) and/or M_L-ary quadrature amplitude modulation (QAM) signals, the phase modulation component can, in complex amplitude, be expressed by, after explicitly denoting data phase d(t), as;

A_{i, k}^{(s i g)} (z, t) \equiv A_{i, k} (z, t) = A^{'}_{i, k} (z, t) \exp {j d (t)}

where, d(t) takes the value of m(π/4) where m = 1, 3, 5, or 7. The corresponding phase-conjugate signal is expressed as

A_{i, k}^{(c o n j)} (z, t) \equiv A_{i, k}^{*} (z, t) = A_{i, k}^{' *} (z, t) \exp {- j d (t)}

Figure 3 is intended to provide a better understanding of the QPSK signal and its phase-conjugate equivalent vector relationship. At the transmitter, the two red symbols correspond to the phase-conjugate of the original symbols, shown in blue. The conjugate pair has reverse sign along the I-axis. When propagating MCF, field evolution is governed by the nonlinear Schrödinger equation. Each PCP experiences NLPS in the same direction by

2 γ_{i} \sum_{k \neq j} {| A_{i, k}^{s i g, c o n j} |}^{2} d z

due to the Kerr effect while the signal envelope broadens due to GVD; Note that the GVD signs are reversed relative to each other because of spectral inversion. Since GVD causes power exchange between the XPM and AM components, phase noise is converted into amplitude noise, the quantity of which increases with CD. Figure 4 shows the simulated constellation maps for signal and conjugate lights under the condition of 5-ch WDM transmission over 16 spans, each of 60 km, with CD values of 6 and 20.5 ps/nm/km. They indicate that the symbol distribution spreads in the direction of the phase rotation while the amplitude noise suggests a chromatic dispersion of 20.5ps/nm/km, because we can see that the radial thickness of the symbol increases when CD is 20.5.

Fig. 3 The vector relationships for phase-conjugate signal pair. A^(sig)(z,t) and A^(conj)(z,t) symbol points for d = π/4 and 3π/4 are plotted for QPSK modulation in (a). Nonlinear phase shift in phase conjugate symbol pair in (b).

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Fig. 4 Simulated constellation maps for phase-conjugate signal pair. Those for chromatic dispersion (CD) values of 6 and 20.5 ps/nm/km are simulated.

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The PCP is combined after weighting. The combined Fourier component is given by

A_{i, k}^{o u t} (Ω) = W_{i, k} (Ω) H_{i, k}^{L} (Ω) {{\tilde{A}}_{i, k} (Ω) H_{i, k}^{N L} (Ω) + {({\tilde{A}}_{i, k}^{*} (Ω) H_{i, k}^{N L} (Ω))}^{*}}

where the Fourier transform of

A_{i, k}^{(s i g), (c o n j)} (z, t)

is

A_{i, k}^{(s i g)} (Ω) \equiv {\tilde{A}}_{i, k} (Ω)

and

A_{i, k}^{(c o n j)} (Ω) \equiv {\tilde{A}}_{i, k}^{*} (Ω)

with the relationships of;

A_{i, k}^{(s i g)} (z, t) = \frac{1}{2 π} \int {\tilde{A}}_{i, k} (Ω) e^{j Ω t} d Ω

and

A_{i, k}^{(c o n j)} (z, t) = \frac{1}{2 π} \int {\tilde{A}}_{i, k}^{*} (Ω) e^{j (- Ω t)} d Ω

Focusing on the nonlinear term in the braces of Eq. (25) and substituting Eq. (19) into it, yields

{\tilde{A}}_{i, k} (Ω) H_{i, k}^{N L} (Ω) + {({\tilde{A}}_{i, k}^{*} (Ω) H_{i, k}^{N L} (Ω))}^{*} = 2 {\tilde{A}}_{i, k} (Ω) \cos {β_{i, k}^{N L} (Ω) z_{r}}

We can automatically obtain the cosine component with double the signal amplitude by the simple PCP combining process. Figure 5 shows the vector relationship to better understand the NLPS cancellation implied by Eq. (27). The constellation maps are indicated as Combined in Fig. 4 for CD values of 6 and 20.5 ps/nm/km. We can see the symbol distributions narrow, compared to the original signal and phase-conjugate symbols described in Fig. 4. Moreover, the smaller CD yields thinner symbol distributions in the radial direction than the larger CD. Decreasing GVD is very effective, assuming that the interference noise induced by four-wave mixing (FMW) can be ignored.

Fig. 5 Combined symbol point, represented in the frequency domain, for phase-conjugate signal pair.

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With this approach, the degree of approximation rises as the nonlinear effective distance shortens. As a consequence, span-by-span GVD compensation (S-by-S) is expected to yield more cancellation than the lumped equivalent at the receiver.

4. WDM signal phase-conjugate pair diversity transmission experiment

The cancellation of nonlinear phase shift was experimentally confirmed and the Q-value gain achieved by MRC was evaluated in the 2-core diversity transmission of mutually phase-conjugated lights through an MCF. The MCFs used had a trench-assisted structure and a cladding diameter of 195 μm and core pitch of 49μm to suppress crosstalk. The measured crosstalk was −65 dB at 1550 nm on average for the 75-km long MCF. The effective core area (Aeff) was about 110 μm² on average. The nonlinear-index coefficient n₂ was about to $32 \times 10^{- 16}$ cm²/W. Attenuation and dispersion at the wavelength of 1550 nm were 0.190–0.199 dB/km, and 20.5–20.8 ps/nm/km, respectively. The experimental setup is shown in Fig. 6. Five optical carriers, with frequency spacing of 50GHz anchored at 193.25THz, emitted from 5 tunable laser sources, were modulated by 25-Gbit/s QPSK code with a 2¹⁵-1 pseudo random bit sequence (PRBS) created by an arbitrary waveform generator (AWGen). The tunable laser spectral linewidths were ~60kHz. They were wavelength-multiplexed, split into two, and input into two IQ modulators. The I- and Q-arms of one were driven by positive logic while Q-arm of the other was driven by negative logic in order to generate the phase-conjugate signal. This phase-conjugatesignal generation scheme follows Eq. (24). The modulated signal traversed the 20km SMF to randomize the bit sequences among the 5 channels. At the output facet, the period difference of 1.5 symbols was set between adjacent channels. The original and phase-conjugated signals were launched into cores #1 and #2 of a 60km 7-core fiber after being amplified by two EDFAs. The launched power to one core was P_in. The signals output from cores #1 and #2 were optically amplified by two EDFAs and launched into #3 and #4; this was repeated for #5 and #6 to create a 3-span transmission system. In S-by-S, DCF’s were inserted in each span; each had insertion loss of about 5dB. Polarization controllers adjusted the polarization state of the original and phase-conjugated signals before the receiver. Demodulation was post-processed offline. The PCP at the frequency of 193.25THz was coherently detected and evaluated. Their CD’s were compensated (weighted) in a DSP. We obtained the Q-value by averaging Q_i and Q_q along the I- and Q-axis, respectively, as in $Q = (Q_{i} + Q_{q}) / 2$ . Q_i and Q_q were estimated from the symbol distributions on the constellation maps by the following formulas; $Q_{i} \equiv 1 / (σ_{i 1} + σ_{i - 1})$ , $Q_{q} \equiv 1 / (σ_{q 1} + σ_{q - 1})$ , where, σ_i1,-1 is the standard deviation ofdistribution for symbol points, 1 and −1, along the I-axis; these points are the mean values normalized by the output voltage corresponding to the constant optical input power level into the receiver circuit. σ_q1,-1 is the standard deviation of symbol points along the Q-axis.

Fig. 6 Experimental setup for WDM phase-conjugate pair diversity transmission. DCF was removed in case of lumped compensation. EDFA; Erbium-Doped Fiber Amplifier, AWGen; Arbitarary Waveform Generator, OBPF; Optical Band Pass Filter, PC; Polarization controller, LO; Local Oscillator, ADC; Analogue-to-Digital Converter.

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The Q value versus total power P^total is shown in Fig. 7 for QPSK coded signals when compensated by S-by-S and Lump. P^total corresponds to Pⁱⁿ = P^total/(M = 1) for single-core transmission and 2 x Pⁱⁿ for 2-core diversity transmission, following the definition in Section 1. Blue indicates the averaged Q value between the original and its conjugate signal single core transmission and green that after MRC of PCP signals. The MRC Q-values, Q_comb, were obtained as follows;

Q_{C o m b} = W_{s i g} Q_{s i g} + W_{c o n j} Q_{c o n j}

where, Q_org and Q_conj denote Q values for the original and its conjugate signals, respectively.W_org and W_conj are inversely proportional to noise power N_k, described in Eq. (13), and satisfy the following relationship;

Fig. 7 Q²-value and its gain vs. fiber launched total power P^total for span-by-span and lump CD compensation. P^total = Pⁱⁿ for single core transmission and P^total = 2xPⁱⁿ for diversity.

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W_{s i g} + W_{c o n j} = 1

Solid lines plot the results for Lump and the dashed line for S-by-S. The S-by-S single transmission Q-value peaked at 24.2 dB and Lump at about 24.5dB at P^total = −2dBm. It droppedfaster than S-by-S as P^total increased. The combined Q-values (green) peaked at 27.9 and 27.1dB for S-by-S and Lump, respectively, when P^total = 1 dBm, and the enhancement over the single transmission Q-value peaks reached 3.7 and 2.6dB, respectively. The Q-value enhancement of 3.7dB means that the transmission distance can be more than doubled, so this approach can significantly raise the bit-rate-distance product, compared to twice the single-core transmission. The black line indicates the difference between single and combined Q-value at the same P^total, that is the Q-value gain. As P^total increases beyond these peak values, the gain in combined Q-value achieved by S-by-S exceeded 10dB at P^total = 10 dBm while Lump achieved 8.9dB. Note that, in Fig. 7, Q-value didn’t increase with P^total in the range of −11 to −2dBm, which is expected to be the linear transmission region. The optical signal-to-noise ratio (OSNR) increased with P^total while the signal level input to the receiver circuit was held constant. We consider that this is mainly because the observed Q-value at the receiver circuit output approached saturation since the thermal noise power remained constant. The cancellation approximation works even in the stronger nonlinearity region. This result verifies that the S-by-S approach cancelled NLPS better than Lump. This is supported by the constellation maps (measured at P^total = 7 and 10dBm) shown in Fig. 8. The combined symbol distributions for S-by-S are clearly narrower than those for Lump. We can also see that Lump compensation yields a larger radial distribution that corresponds to the amplitude noise component. The MRC process can’t cancel this component.

Fig. 8 Measured constellation maps for original and its phase conjugate signal pair and their combination. Fiber launched power P^total was 7 and 10dBm.

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The nonlinear phase-shift cancellation for smaller GVD was also confirmed by simulation. Figure 9 shows the Q-value (blue and green) and its gain (black) versus P^total for CD = 6 and 20.5 ps/nm/km. The Q value for CD = 6 ps/nm/km in single core transmission peaked at 18.4dB at −5dBm while that for CD = 20.5 peaked at 18dB at −3dBm. The MRC Q-value for CD = 6 (solid green) peaked at 24dB at P^total = 0dBm, showing a Q-value gain of 5.6dB. The Q-value gain strengthens with nonlinearity and reaches 14dB when CD = 6 ps/nm/km. We can see that smaller CD is more effective than larger CD.

Fig. 9 Q²-value and its gain vs. fiber launched total power P^total for CD = 6 and 20.5 ps/nm/km. P^total = Pⁱⁿ for single core transmission and P^total = 2xPⁱⁿ for diversity.

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Conclusion

We proposed an MRC scheme for WDM PCP diversity transmission to cancel nonlinear phase-shift. A transfer function approximation for nonlinear phase noise cancellation was formulated and it was shown, with the help of numerical calculations, that span-by-span CD compensation should yield more cancellation than its lumped equivalent at the receiver. Assuming that the interference noise induced by FWM can be ignored, Q-value was enhanced much more at smaller CD than at large ones. The proposal was validated in a 2-core diversity 5 channel WDM transmission experiment over 3-spans of 60km MCF using 25 Gbit/s-QPSK PCP. The Q-value enhancement by MRC reached 3dB at the peak, yielding superior bitrate-distance product and optical power density limit, compared to twice single-core transmission.

Acknowledgment

Part of this research uses results from research commissioned by the National Institute of Information and Communications Technology (NICT) entitled “Research on innovative infrastructure of optical communications”.

References and links

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Optical diversity transmission using WDM signal and phase-conjugate lights through multi-core fiber

Abstract

1. Introduction

2. Optical diversity transmission and maximum-ratio combining

3. Transfer function approximation

3.1 L-span repeat WDM transmission

3. 2 Nonlinear phase-shift cancellation by WDM phase-conjugate light pair

4. WDM signal phase-conjugate pair diversity transmission experiment

Conclusion

Acknowledgment

References and links

Cited By

Figures (9)

Equations (30)

Optics Express