Wavelength/polarization division multiplexing (WDM/PDM) schemes are standardly employed in trunk transmission systems, for which fiber nonlinearity is a serious problem. Generally, the signal transmission characteristics suffering from fiber nonlinearity have been analyzed using the nonlinear Schrödinger equation or the Manakov equation, which was originally presented for single-channel transmission. Therefore, when they are applied to WDM systems, the polarization states of each wavelength light are assumed to be identical, which is not fit to long-haul transmissions wherein the polarization states of WDM signals randomly and independently vary during the fiber transmission. Considering such conditions in WDM/PDM transmission systems, this paper presents a nonlinear wave equation for WDM/PDM signal transmission, which describes the signal behavior in systems where the polarization states of each WDM light are fully random and independent. A formula modified from the Manakov equation is derived using the polarization state coordinate of each signal light in WDM/PDM systems.
1. Introduction
In optical trunk transmission systems, wavelength and polarization division multiplexing (WDM/PDM) schemes are standardly employed to achieve large transmission capacities. In such transmission systems, fiber nonlinearity is a serious impairment factor that restricts the system performance. Studies on the fiber nonlinearity effects on optical transmission systems have been intensively conducted to clarify and improve the signal transmission performance limited by fiber nonlinearity.
The nonlinear Schrödinger equation has been commonly used for the theoretical evaluation of the nonlinear signal transmission characteristics. However, it is a scalar equation describing the light behavior in one polarization mode, and is not suitable for polarization division multiplexed (PDM) signals. The Manakov equation is applicable for polarization-state-related nonlinearity, which is a pair of coupled-equations extended from the nonlinear Schrödinger equation, and was originally presented for analyzing the cross-phase modulation between two orthogonally polarized components of one signal light in soliton transmission [1,2]. Several studies have applied the Manakov equation to WDM systems, investigating the cross-polarization modulation effect between WDM signal lights [3,4], where one polarization mode for one wavelength light was assumed, not PDM signals. One study investigated WDM/PDM systems, using the Manakov equation [5], assuming that the polarization states of WDM signals are all aligned. However, in fiber transmission systems, the polarization states of different wavelength lights are generally independent of each other, and vary differently for each wavelength during the propagation through a transmission fiber. The condition of the same polarization state for any WDM signal is a simplified model for the analysis, or may be a worst-case scenario.
Considering the above situation, this study presents a nonlinear wave equation for WDM/PDM fiber transmission systems, where the polarization states of the PDM signals are fully random for each wavelength. A nonlinear wave equation modified from the Manakov equation to describe WDM signal lights is derived by introducing the polarization coordinate matched to a polarization-multiplexed signal light for each wavelength.
2. Nonlinear polarization
For the x component of light propagating in the z direction through an isotropic medium, such as a glass material, the nonlinear polarization is expressed as [6]
(1)$$P_x^{\textrm{NL}} = 3{c_{xxxx}}{E_x}{E_x}E_x^\ast{+} 3{c_{xxyy}}{E_x}{E_y}E_y^\ast{+} 3{c_{xyxy}}{E_y}{E_x}E_y^\ast{+} 3{c_{xyyx}}{E_y}{E_y}E_x^\ast , $$
where cxijk (i, j, k, = {x, y}) denotes a tensor component of the third-order nonlinear susceptibility with the relationship of cxxyy = cxyxy = cxyyx = cxxxx/3, and Ex,y denotes the light amplitude along the x or y axis perpendicular to the propagation direction. Equation (1) indicates the locally induced nonlinear polarization. For light propagating through a length of a birefringent medium, the last term in Eq. (1) is phase-mismatched with the x-polarized light, and its effect is negligible compared with the phase-matched terms. Excluding this term, Eq. (1) can be rewritten as follows [3]: (2a)$$P_x^{\textrm{NL}} {=}\; \gamma (|{E_x}{|^2} + |{E_y}{|^2}){E_x} - \frac{\gamma }{3}|{E_y}{|^2}{E_x}. $$
where the nonlinear susceptibility is replaced with the nonlinear parameter γ, which includes the mode confinement effect in a waveguide and is usually used for fiber nonlinearity. Similarly, the nonlinear polarization of the y component is expressed as follows: (2b)$$P_y^{\textrm{NL}} = \gamma (|{E_x}{|^2} + |{E_y}{|^2}){E_y} - \frac{\gamma }{3}|{E_x}{|^2}{E_y}. $$
The above equations can be summarized in the vector representation as
(3)$${{\mathbf P}^{\textrm{NL}}} = \left( {\begin{array}{{c}} {P_x^{\textrm{NL}}}\\ {P_y^{\textrm{NL}}} \end{array}} \right) = P_x^{\textrm{NL}}{{\mathbf e}_x} + P_y^{\textrm{NL}}{{\mathbf e}_y}, $$
where ex = (1, 0)T and ey = (0, 1)T indicate the unit vectors along the x and y axes, respectively.In Eq. (3), the nonlinear polarization is expressed in the x-y coordinate, which is fixed to the cross-section of a transmission fiber. In this work, we consider polarization-multiplexed signals whose polarization states are not aligned along the x and y axes in general. To treat such a situation, we introduce the complex unit vectors (or basis states) that represent the polarization states of polarization multiplexed signal lights, denoted as e+ and e-. These unit vectors are generally expressed by the unit vectors of the x-y coordinate, as follows:
(4a)$${{\mathbf e}_ + } = {{\mathbf e}_x}\cos \phi + {{\mathbf e}_y}{e^{i\Delta }}\sin \phi, $$
(4b)$${{\mathbf e}_ - } ={-} {{\mathbf e}_x}{e^{ - i\Delta }}\sin \phi + {{\mathbf e}_y}\cos \phi, $$
where ϕ and Δ are parameters indicating the relationship between the signal polarization coordinate and the x-y coordinate fixed to the fiber medium. The above vectors satisfy the orthogonality of ${\mathbf e}_ + ^\ast{\cdot} {{\mathbf e}_ - } = 0$ and the normalization condition of |e+| = |e-| = 1.Using the above notations, the field amplitudes of the multiplexed signal polarization states and those along the x and y axes are related as follows:
(5)$$\begin{aligned} {\mathbf E} &= {E_x}{{\mathbf e}_x} + {E_y}{{\mathbf e}_y} = {E_ + }{{\mathbf e}_ + } + {E_ - }{{\mathbf e}_ - } = {E_ + }({{\mathbf e}_x}\cos \phi + {{\mathbf e}_y}{e^{i\Delta }}\sin \phi ) + {E_ - }( - {{\mathbf e}_x}{e^{ - i\Delta }}\sin \phi + {{\mathbf e}_y}\cos \phi )\\& = ({E_ + }\cos \phi - {E_ - }{e^{ - i\Delta }}\sin \phi ){{\mathbf e}_x} + ({E_ + }{e^{i\Delta }}\sin \phi + {E_ - }\cos \phi ){{\mathbf e}_y}, \end{aligned}$$
and subsequently, (6a)$${E_x} = {E_ + }\cos \phi - {E_ - }{e^{ - i\Delta }}\sin \phi, $$
(6b)$${E_y} = {E_ + }{e^{i\Delta }}\sin \phi + {E_ - }\cos \phi, $$
where E+ and E- are the amplitudes of the signal polarization states, respectively, and Ex and Ey are the those along the x and y axes, respectively. From Eq. (6), the following relationship is obtained: (7a)$${E_ + } = {E_x}\cos \phi + {E_y}{e^{ - i\Delta }}\sin \phi, $$
(7b)$${E_ - } ={-} {E_x}{e^{i\Delta }}\sin \phi + {E_y}\cos \phi. $$
The above relationship is also satisfied for the nonlinear polarization field, and the nonlinear polarization in the signal polarization states, P + NL and P-NL, are expressed as follows:
(8a)$$P_ + ^{\textrm{NL}} = P_x^{\textrm{NL}}\cos \phi + P_y^{\textrm{NL}}{e^{ - i\Delta }}\sin \phi, $$
(8b)$$P_ - ^{\textrm{NL}} ={-} P_x^{\textrm{NL}}{e^{i\Delta }}\sin \phi + P_y^{\textrm{NL}}\cos \phi. $$
Substituting Eqs. (2) and (6) into (8), the nonlinear polarization can be expressed in the signal polarization coordinate, which is performed for wavelength/polarization multiplexed signal lights in the next section. Note that the light amplitudes in the above represent not only continuous waves but also modulated signals in fiber communication systems in general.3. Wavelength/polarization multiplexed signals
During the propagation through a long fiber line, the polarization states of different wavelength lights cannot be uniformly aligned, as indicated in [Appendix A]. Therefore, the polarization states of each signal light are random in long-haul WDM/PDM transmission systems. We derive a nonlinear wave equation that describes the signal behavior in such systems, using the formulas presented in the previous section.
To treat various polarization states, we introduce the unit vectors of the signal polarization states of the kth wavelength light as follows:
(9a)$${{\mathbf e}_{k + }} = {{\mathbf e}_x}\cos {\phi _k} + {{\mathbf e}_y}{e^{i{\Delta _k}}}\sin {\phi _k}, $$
(9b)$${{\mathbf e}_{k - }} ={-} {{\mathbf e}_x}{e^{ - i{\Delta _k}}}\sin {\phi _k} + {{\mathbf e}_y}\cos {\phi _k}, $$
where ϕk and Δk indicate the signal polarization state of the kth wavelength light at a local point along the fiber length with respect to the x-y coordinate fixed to the fiber medium. Similar to Eqs. (6) and (7), the relationship between the field amplitudes of the kth wavelength in the x-y coordinate and those in the signal polarization state coordinate can be expressed as follows: (10a)$$E_x^{(k)} = {E_{k + }}\cos {\phi _k} - {E_{k - }}{e^{ - i{\Delta _k}}}\sin {\phi _k}, $$
(10b)$$E_y^{(k)} = {E_{k + }}{e^{i{\Delta _k}}}\sin {\phi _k} + {E_{k - }}\cos {\phi _k}. $$
(11a)$${E_{k + }} = E_x^{(k)}\cos {\phi _k} + E_y^{(k)}{e^{ - i{\Delta _k}}}\sin {\phi _k}, $$
(11b)$${E_{k - }} ={-} E_x^{(k)}{e^{i{\Delta _k}}}\sin {\phi _k} + E_y^{(k)}\cos {\phi _k}. $$
where {Ex(k), Ey(k)} and {Ek+, Ek-} are the amplitudes of the kth wavelength in the x-y coordinate and those in the signal polarization coordinate, respectively.Using the above notations, the x and y components of the wavelength multiplexed lights are expressed as
(12a)$${E_x} = \sum\limits_k {E_x^{(k)}{e^{ik2\pi \Delta ft}}} = \sum\limits_k {({E_{k + }}\cos {\phi _k} - {E_{k - }}{e^{ - i{\Delta _k}}}\sin {\phi _k}){e^{ik2\pi \Delta ft}}}, $$
(12b)$${E_y} = \sum\limits_k {E_y^{(k)}{e^{ik2\pi \Delta ft}}} = \sum\limits_k {({E_{k + }}{e^{i{\Delta _k}}}\sin {\phi _k} + {E_{k - }}\cos {\phi _k}){e^{ik2\pi \Delta ft}}}, $$
and their intensities are expressed as (13a)$$\begin{array}{l} |{E_x}{|^2} = {\left|{\sum\limits_k {({E_{k + }}\cos {\phi_k} - {E_{k - }}{e^{ - i{\Delta _k}}}\sin {\phi_k}){e^{ik2\pi \Delta ft}}} } \right|^2} = \sum\limits_k {|{E_{k + }}\cos {\phi _k} - {E_{k - }}{e^{ - i{\Delta _k}}}\sin {\phi _k}{|^2}} \\ = \sum\limits_k {\{ |{E_{k + }}{|^2}{{\cos }^2}{\phi _k} + |{E_{k - }}{|^2}{{\sin }^2}{\phi _k} - ({E_{k + }}E_{k - }^\ast {e^{i{\Delta _k}}} + E_{k + }^\ast {E_{k - }}{e^{ - i{\Delta _k}}})\cos {\phi _k}\sin {\phi _k}\} } \end{array}$$
(13b)$$|{E_y}{|^2} = \sum\limits_k {\{ |{E_{k + }}{|^2}{{\sin }^2}{\phi _k} + |{E_{k - }}{|^2}{{\cos }^2}{\phi _k} + ({E_{k + }}E_{k - }^\ast {e^{i{\Delta _k}}} + E_{k + }^\ast {E_{k - }}{e^{ - i{\Delta _k}}})\cos {\phi _k}\sin {\phi _k}\} }, $$
where Δf is the frequency spacing between WDM channels. In Eq. (13), the beat frequency components, such as (k – k’)Δf for k ≠ k’, are neglected. These components locally induce the nonlinear polarization at a target signal frequency, fs, when coupled with light at fs ± (k – k’)Δf, through the four-wave mixing process. However, the effect of these nonlinear polarization terms is negligible owing to the phase-mismatch resulting from the chromatic dispersion, provided that the transmission line is composed of normal-dispersion fibers in which the chromatic dispersion is typically –17 ps/km-nm for 1.55-µm wavelength light. Therefore, these terms are neglected in Eq. (13). From Eq. (13), the total intensity is expressed as follows: (14)$$|{E_x}{|^2} + |{E_y}{|^2} = \sum\limits_k {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})}. $$
Similar to Eq. (13a), the nonlinear polarization in the polarization state es+ can be expressed as follows:
(15)$$P_{s + }^{\textrm{NL}} = P_{s(x)}^{\textrm{NL}}\cos {\phi _s} + P_{s(y)}^{\textrm{NL}}{e^{ - i{\Delta _s}}}\sin {\phi _s}. $$
$P_{s(x)}^{\textrm{NL}}$ and $P_{s(y)}^{\textrm{NL}}$ in this expression are the nonlinear polarizations of the sth wavelength along the x and y axes, respectively, which are expressed from Eqs. (2), (10), (13), and (14) as (16a)$$\begin{aligned} P_{s(x)}^{\textrm{NL}} &= (|{E_x}{|^2} + |{E_y}{|^2})E_x^{(s)} - \frac{\gamma }{3}|{E_y}{|^2}E_x^{(s)}\\ = &\; ({E_{s + }}\cos {\phi _s} - {E_{s - }}{e^{ - i{\Delta _s}}}\sin {\phi _s})\sum\limits_k {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})} \\& - \frac{\gamma }{3}({E_{s + }}\cos {\phi _s} - {E_{s - }}{e^{ - i{\Delta _s}}}\sin {\phi _s})\\& \;\;\times \sum\limits_k {\{ |{E_{k + }}{|^2}{{\sin }^2}{\phi _k} + |{E_{k - }}{|^2}{{\cos }^2}{\phi _k} + ({E_{k + }}E_{k - }^\ast {e^{i{\Delta _k}}} + E_{k + }^\ast {E_{k - }}{e^{ - i{\Delta _k}}})\cos {\phi _k}\sin {\phi _k}\} } \end{aligned}$$
(16b)$$\begin{aligned} P_{s(y)}^{\textrm{NL}} &= (|{E_x}{|^2} + |{E_y}{|^2})E_y^{(s)} - \frac{\gamma }{3}|{E_x}{|^2}E_y^{(s)}\\ = &\;({E_{s + }}{e^{i{\Delta _s}}}\sin {\phi _s} + {E_{s - }}\cos {\phi _s})\sum\limits_k {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})} \\ &- \frac{\gamma }{3}({E_{s + }}{e^{i{\Delta _s}}}\sin {\phi _s} + {E_{s - }}\cos {\phi _s})\\ &\;\;\times \sum\limits_k {\{ |{E_{k + }}{|^2}{{\cos }^2}{\phi _k} + |{E_{k - }}{|^2}{{\sin }^2}{\phi _k} - ({E_{k + }}E_{k - }^\ast {e^{i{\Delta _k}}} + E_{k + }^\ast {E_{k - }}{e^{ - i{\Delta _k}}})\cos {\phi _k}\sin {\phi _k}\} } \end{aligned}$$
Substituting Eqs. (16) into (15) and expanding the formula, the following expression can be derived: (17)$$\begin{aligned} P_{s + }^{\textrm{NL}} =&\; \gamma (|{E_{s + }}{|^2} + |{E_{s - }}{|^2}){E_{s + }} + \gamma {E_{s + }}\sum\limits_{k \ne s} {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})} \\ &- \frac{\gamma }{3}\left[ {\frac{1}{2}|{E_{s + }}{|^2}{E_{s + }}{{\sin }^2}2{\phi_s} + \frac{1}{2}|{E_{s - }}{|^2}{E_{s + }}(1 + {{\cos }^2}2{\phi_s})} \right.\\ &\;\;\;\;+ \frac{1}{4}(|{E_{s + }}{|^2}{E_{s - }}{e^{ - i{\Delta _s}}} + {E_{s + }}^2{E_{s - }}^\ast {e^{i{\Delta _s}}})\sin 4{\phi _s} + \frac{1}{4}(|{E_{s + }}{|^2} - |{E_{s - }}{|^2}){E_{s - }}{e^{ - i{\Delta _s}}}\sin 4{\phi _s}\\ &\;\;\;\;- \frac{1}{2}({E_{s + }}^\ast {E_{s - }}^2{e^{ - i2{\Delta _s}}} + {E_{s + }}|{E_{s - }}{|^2}){\sin ^2}2{\phi _s}\\ &\;\;\;\;+ ({E_{s + }}{\cos ^2}{\phi _s} - \frac{1}{2}{E_{s - }}{e^{ - i{\Delta _s}}}\sin 2{\phi _s})\sum\limits_{k \ne s} {(|{E_{k + }}{|^2}{{\sin }^2}{\phi _k} + |{E_{k - }}{|^2}{{\cos }^2}{\phi _k})} \\ &\;\;\;\;+ ({E_{s + }}{\sin ^2}{\phi _s} + \frac{1}{2}{E_{s - }}{e^{ - i{\Delta _s}}}\sin 2{\phi _s})\sum\limits_{k \ne s} {(|{E_{k + }}{|^2}{{\cos }^2}{\phi _k} + |{E_{k - }}{|^2}{{\sin }^2}{\phi _k})} \\&\;\;\;\; \left. { + \frac{1}{2}({E_{s + }}\cos 2{\phi_s} - {E_{s - }}{e^{ - i{\Delta _s}}}\sin 2{\phi_s})\sum\limits_{k \ne s} {({E_{k + }}^\ast {E_{k - }}{e^{ - i{\Delta _k}}} + {E_{k + }}{E_{k - }}^\ast {e^{i{\Delta _k}}})\sin 2{\phi_k}} } \right] \end{aligned}$$
In this expression, parameters ϕk and Δk, that indicate the polarization state of the signal light with respect to the x-y coordinate fixed to the fiber medium, randomly vary during the fiber transmission. Therefore, we average the terms related to these parameters, such that < cos2(2ϕ)> = 1/3, <sin2(2ϕ)> = 2/3, <e ± iΔsin(4ϕ)> = <e-iΔsin(2ϕ)> = 0, and < cos2ϕ> = <sin2ϕ> = 1/2 [Appendix B]. Applying this averaging, Eq. (17) is rewritten as (18a)$$P_{s + }^{\textrm{NL}} = \frac{8}{9}\gamma \left\{ {|{E_{s + }}{|^2} + |{E_{s - }}{|^2} + \frac{{15}}{{16}}\sum\limits_{k \ne s} {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})} } \right\}{E_{s + }}. $$
Similarly, the nonlinear polarization in the polarization state es- is expressed as (18b)$$P_{s - }^{\textrm{NL}} = \frac{8}{9}\gamma \left\{ {|{E_{s + }}{|^2} + |{E_{s - }}{|^2} + \frac{{15}}{{16}}\sum\limits_{k \ne s} {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})} } \right\}{E_{s - }}. $$
In the above equations, the first, second, and third terms represent the self-phase modulation, the cross-phase modulations from the orthogonally polarized light of the same wavelength, and the cross-phase modulation from different wavelength lights, respectively.By applying the above nonlinear polarization to a conventional wave equation for polarization-multiplexed light of the sth wavelength, the following nonlinear wave equations can be obtained:
(19a)$$\begin{aligned} \frac{{\partial {E_{s + }}}}{{\partial z}} &+ \frac{\alpha }{2}{E_{s + }} + i\frac{{\beta _{s + }^{(2)}}}{2}\frac{{{\partial ^2}{E_{s + }}}}{{\partial {t^2}}}\\& = i\frac{8}{9}\gamma \left\{ {|{E_{s + }}{|^2} + |{E_{s - }}{|^2} + \frac{{15}}{{16}}\sum\limits_{k \ne s} {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})} } \right\}{E_{s + }} \end{aligned}$$
(19b)$$\begin{aligned} \frac{{\partial {E_{s - }}}}{{\partial z}} &+ \frac{\alpha }{2}{E_{s - }} + i\frac{{\beta _{s - }^{(2)}}}{2}\frac{{{\partial ^2}{E_{s - }}}}{{\partial {t^2}}}\\& = i\frac{8}{9}\gamma \left\{ {|{E_{s + }}{|^2} + |{E_{s - }}{|^2} + \frac{{15}}{{16}}\sum\limits_{k \ne s} {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})} } \right\}{E_{s - }} \end{aligned},$$
where α denotes the fiber attenuation coefficient, and βs + (2) and βs-(2) are the group velocity dispersions for the PDM signal lights of the sth wavelength, respectively. The signal light behavior in WDM/PDM fiber transmission systems under the condition that the polarization states are random and independent of other wavelength lights can be described by the above nonlinear wave equations.The derived formula shown in Eq. (19) is similar to the Manakov equation, that is expressed as [4]
(20)$$\frac{{\partial {\mathbf E}}}{{\partial z}} + \frac{\alpha }{2}{\mathbf E} + i\frac{{{\beta ^{(2)}}}}{2}\frac{{{\partial ^2}{{\mathbf E}}}}{{\partial {t^2}}} = i\frac{8}{9}\gamma |{\mathbf E}{|^2}{\mathbf E}.$$
Conventionally, when applying this formula to WDM/PDM systems, the field amplitude E = (E+, E-)T is regarded as E+ = ΣkEk + exp(ik2πΔft) and E- = ΣkEk-exp(ik2πΔft), which are then substituted to Eq. (20) [5]. In this conventional treatment, the nonlinear polarization in the polarization state e+ at the sth wavelength is expressed as (21)$$P_{s + }^{\textrm{NL}} = \frac{8}{9}\gamma \left\{ {\sum\limits_k {(|{E_{k + }}{|^2} + |{E_{k - }}{|^2})} } \right\}{E_{s + }}, $$
provided that the beat intensity oscillations are negligible because of the phase-mismatch as described below Eq. (13). The difference in our formula [Eq. (18a)] and the Manakov equation based treatment [Eq. (21)] is the cross-phase modulation effect from different wavelength lights, such that factor 15/16 is attached in our formula whereas it is not in the Manakov equation. This difference originates from that the terms for the concerned signal wavelength and other wavelengths are separately treated as in Eq. (17) considering the difference in the polarization state at different wavelengths.Another difference in our formula and the Manakov equation is that the group velocity dispersions for each signal light are exhibited in Eq. (19) whereas one group velocity dispersion is commonly used in Eq. (20). This is because the Manakov equation was originally presented for one signal light [1,2]. Therefore, our formula would properly describe the nonlinear transmission characteristics of WDM/PDM systems concerning the group velocity dispersion for each signal light.
4. Summary
This paper presented a nonlinear wave equation for WDM/PDM fiber transmission systems, where the polarization states of multiplexed lights vary randomly and independently for each wavelength. A formula, modified from the Manakov equation to describe WDM lights with random polarization states, was derived by introducing the signal polarization coordinate and averaging the signal polarization state with respect to the x-y coordinate fixed to the fiber medium. The signal behavior in long-haul WDM/PDM transmission systems can be analyzed with the presented equation.
Appendix A
This appendix introduces an experiment that we carried out to indicate that the polarization states of WDM signals are not aligned during long-haul fiber transmission. Figure 1 shows the setup. The laser light from a wavelength-tunable LD was launched onto a single-mode fiber with a power level of 0 dBm, the output from which passed through a polarization beam splitter (PBS) followed by power meters. The polarization state of the tunable LD output was unchanged for any wavelength. With this setup, the output powers at the PBS outputs were measured while changing the LD wavelength.
The results are shown in Fig. 2, where the power ratio of Px/(Px + Py) is plotted with Px and Py being the two output powers from the PBS, respectively. Two fiber lengths were examined: 50 km (a) and 25 km (b). The power ratio was observed to vary with the LD wavelength, indicating that the polarization state after the fiber transmission was different depending on the wavelength, even though the fiber input state (or the LD output state) was identical. The wavelength difference between the peak and bottom of the power ratio, at which the polarization states were completely orthogonal, was 20 nm for the 50-km fiber and 40 nm for the 25-km fiber. This result suggests that the polarization states of two signal lights with a wavelength difference of a couple of nanometers could become orthogonal to each other after a 500-km fiber transmission. Subsequently, long-haul transmission would fully randomize the polarization states of each wavelength light. Although this experiment used a continuous-wave light, the result is applicable to modulated light because how the polarization state varies is irrespective of whether the light is modulated or not.
Appendix B
In this appendix, we average the terms related to the polarization state in Eq. (17). First, the signal polarization state is expressed in the x-y coordinate, or as a Jones vector, as follows:
(22)$${{\mathbf e}_ + } = \left( {\begin{array}{{c}} {\cos \phi }\\ {{e^{i\Delta }}\sin \phi } \end{array}} \right). $$
This state is also expressed by a Stokes vector as
(23)$$\left( {\begin{array}{{c}} {{S_1}}\\ {{S_2}}\\ {{S_3}} \end{array}} \right) = \left( {\begin{array}{{c}} {{{\cos }^2}\phi - {{\sin }^2}\phi }\\ {2\cos \phi \sin \phi \cos \Delta }\\ {2\cos \phi \sin \phi \sin \Delta } \end{array}} \right) = \left( {\begin{array}{{c}} {\cos 2\phi }\\ {\sin 2\phi \cos \Delta }\\ {\sin 2\phi \sin \Delta } \end{array}} \right). $$
This polarization state is schematically indicated by a point on a Poincaré sphere in the Stokes vector space, as illustrated in Fig. 3, where 2ϕ = ψ is the angle between the S1 axis and the Stokes vector and Δ is the angle between the S2 axis and the Stokes vector projected onto the S2-S3 plane. For fully random polarization states, their Stokes vectors are uniformly distributed on the Poincaré sphere. Subsequently, the mean value of a stochastic variable ξ, that is a function of ψ and Δ as ξ = f(ψ, Δ) in general, is evaluated as
(24)$$< \xi > = \frac{1}{{4\pi }}\int\limits_0^\pi {\sin \psi d\psi \int\limits_0^{2\pi } {f(\psi ,\Delta )d\Delta } }. $$
Using this formula, the mean values of the terms related to the signal polarization state in Eq. (
17) are obtained as follows:
(25)$$< {\cos ^2}2\phi > = \frac{1}{{4\pi }}\int\limits_0^\pi {{{\cos }^2}\psi \sin \psi d\psi \int\limits_0^{2\pi } {d\Delta } } = \frac{1}{3}, $$
(26)$$< {e^{ {\pm} i\Delta }}\sin 4\phi > = \frac{1}{{4\pi }}\int\limits_0^\pi {\sin 2\psi \sin \psi d\psi \int\limits_0^{2\pi } {{e^{ {\pm} i\Delta }}d\Delta } } = 0, $$
(27)$$< {\cos ^2}\phi > = \frac{1}{{4\pi }}\int\limits_0^\pi {\frac{1}{2}(1 + \cos \chi )\sin \psi d\psi \int\limits_0^{2\pi } {d\Delta } } = \frac{1}{2}, $$
(28)$$< \cos 2\phi > = \frac{1}{{4\pi }}\int\limits_0^\pi {\cos \psi \sin \psi d\psi \int\limits_0^{2\pi } {d\Delta } } = 0, $$
(29)$$< {e^{ - i\Delta }}\sin 2\phi > = \frac{1}{{4\pi }}\int\limits_0^\pi {{{\sin }^2}(\psi )d\psi \int\limits_0^{2\pi } {{e^{ - i\Delta }}d\Delta } } = 0, $$
and < sin
2(2
ϕ)> = <1 – cos
2(2
ϕ) > = 2/3 and < sin
2(
ϕ)> = <1 – cos
2(
ϕ) > = 1/2. These mean values are applied to Eq. (
17).
Disclosures
The authors declare 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 authors upon reasonable request.
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