## Abstract

By assuming the nonlinear noise as a signal-independent circular Gaussian noise, a typical case in non-dispersion managed links with coherent multilevel modulation formats, we provide several analytical properties of a new quality parameter – playing the role of the signal to noise ratio (SNR) at the sampling gate in the coherent receiver – which carry over to the Q-factor versus power (or “bell”) curves. We show that the maximum Q is reached at an optimal power, the nonlinear threshold, at which the amplified spontaneous emission (ASE) noise power is twice the nonlinear noise power, and the SNR penalty with respect to linear propagation is
$10\text{Log}\left(\frac{3}{2}\right)\hspace{0.17em}\simeq \hspace{0.17em}1.76\hspace{0.17em}\text{dB},$, although the Q-penalty is somewhat larger and increases at lower Q-factors, as we verify for the polarization-division multiplexing quadrature phase shift keying (PDM-QPSK) format. As we vary the ASE power, the maxima of the SNR vs. power curves are shown to slide along a straight-line with slope ≃−2 dB/dB. A similar behavior is followed by the Q-factor maxima, although for PDM-QPSK the local slope is around −2.7 dB/dB for Q-values of practical interest.

© 2011 OSA

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### Equations (13)

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(1)
$$r\hspace{0.17em}=\hspace{0.17em}s\sqrt{P}\hspace{0.17em}+\hspace{0.17em}{n}_{L}\hspace{0.17em}+\hspace{0.17em}{n}_{\mathit{\text{NL}}}$$
(2)
$${N}_{\mathit{\text{NL}}}\hspace{0.17em}=\hspace{0.17em}\mathit{\text{Var}}[{n}_{\mathit{\text{NL}}}]\hspace{0.17em}=\hspace{0.17em}{a}_{\mathit{\text{NL}}}{P}^{3}\hspace{0.17em}[\text{W}]$$
(3)
$$S\hspace{0.17em}=\hspace{0.17em}\frac{P}{{N}_{A}\hspace{0.17em}+\hspace{0.17em}{a}_{\mathit{\text{NL}}}{P}^{3}}.$$
(4)
$${S}_{L,\mathit{\text{dB}}}\hspace{0.17em}\triangleq \hspace{0.17em}{P}_{\mathit{\text{dB}}}\hspace{0.17em}-\hspace{0.17em}{N}_{A,\mathit{\text{dB}}}\hspace{0.17em}\text{if}\hspace{0.17em}P\hspace{0.17em}\ll \hspace{0.17em}{P}_{B}$$
(5)
$${S}_{R,\mathit{\text{dB}}}\hspace{0.17em}\triangleq \hspace{0.17em}-2{P}_{dB}\hspace{0.17em}-\hspace{0.17em}{a}_{\mathit{\text{NL}},\mathit{\text{dB}}}\hspace{0.17em}\text{if}\hspace{0.17em}P\hspace{0.17em}\gg \hspace{0.17em}{P}_{B}$$
(6)
$$S\hspace{0.17em}=\hspace{0.17em}\frac{{S}_{L}}{1\hspace{0.17em}+\hspace{0.17em}\frac{{a}_{\mathit{\text{NL}}}{P}^{3}}{{N}_{A}}}$$
(7)
$$\mathit{\text{SP}}\hspace{0.17em}=\hspace{0.17em}1\hspace{0.17em}+\hspace{0.17em}\frac{{a}_{\mathit{\text{NL}}}{P}^{3}}{{N}_{A}}$$
(8)
$${N}_{A}\hspace{0.17em}=\hspace{0.17em}2({a}_{\mathit{\text{NL}}}{P}^{3})$$
(9)
$${P}_{\mathit{\text{NLT}}}\hspace{0.17em}=\hspace{0.17em}{\left(\frac{{N}_{A}}{2{a}_{\mathit{\text{NL}}}}\right)}^{\frac{1}{3}}$$
(10)
$${S}_{\mathit{\text{NLT}}}\hspace{0.17em}=\hspace{0.17em}\frac{{P}_{\mathit{\text{NLT}}}}{\frac{3}{2}{N}_{A}}\hspace{0.17em}=\hspace{0.17em}{\left({3}^{3}\hspace{0.17em}{a}_{\mathit{\text{NL}}}{(\frac{{N}_{A}}{2})}^{2}\right)}^{-\frac{1}{3}}\hspace{0.17em}.$$
(11)
$$Q\hspace{0.17em}=\hspace{0.17em}\sqrt{2}{\text{erfc}}^{-1}\hspace{0.17em}(2\mathit{\text{BER}}(S))$$
(12)
$${Q}_{\mathit{\text{dB}}}^{2}\hspace{0.17em}=\hspace{0.17em}-A\hspace{0.17em}\cdot \hspace{0.17em}\mathit{\text{OSN}}{R}_{\mathit{\text{dB}}}^{2}\hspace{0.17em}+\hspace{0.17em}B\hspace{0.17em}\cdot \hspace{0.17em}\mathit{\text{OSN}}{R}_{\mathit{\text{dB}}}\hspace{0.17em}-\hspace{0.17em}C$$
(13)
$$Q{P}_{\mathit{\text{dB}}}\hspace{0.17em}=\hspace{0.17em}S{P}_{\mathit{\text{dB}}}\hspace{0.17em}\cdot \hspace{0.17em}[B\hspace{0.17em}-\hspace{0.17em}A\hspace{0.17em}\cdot \hspace{0.17em}(S{P}_{\mathit{\text{dB}}}\hspace{0.17em}+\hspace{0.17em}2\hspace{0.17em}\cdot \hspace{0.17em}({S}_{\mathit{\text{dB}}}\hspace{0.17em}+\hspace{0.17em}b)]$$