Polarization-division-multiplexing (PDM) is widely used in fiber transmission systems, wherein fiber nonlinearity is a crucial issue to be considered. Conventionally, the Manakov equation has been used to analyze the nonlinear propagation properties of PDM signal lights, which describes the nonlinear interaction between orthogonally polarized lights. However, because the polarization states of PDM signals are not necessarily orthogonal in transmission systems with polarization-dependent loss (PDL), it is not certain if the Manakov equation is applicable to such systems. Therefore, this study presents a wave equation that describes the nonlinear interaction between non-orthogonally polarized PDM signal lights. We derive a formula that considers the nonorthogonality resulting from PDL. Calculation based on the formula is carried out, the result of which shows that the nonlinear wave equation assuming orthogonal PDM signals results in negligible errors in effect when treating nonorthogonal PDM signals.
1. Introduction
Polarization multiplexing schemes are widely used in optical fiber communications to double the transmission capacity. However, fiber nonlinearity is a crucial issue in these systems. The coupled Schrödinger equation or the Manakov equation, which is a pair of nonlinear wave equations describing the behaviors of orthogonally polarized signal lights, is typically used to theoretically analyze the nonlinear effect on polarization-division-multiplexed (PDM) signals.
The Manakov equation was originally presented to describe the nonlinear interaction between two orthogonally polarized components of one signal light in a fiber with slight birefringence [1,2]. Thereafter, it has been applied to PDM signal lights since PDM systems started to be developed [3,4], based on the condition that the polarization states of PDM signals are orthogonal to each other.
However, the condition of orthogonality between PDM signals is not necessarily satisfied in transmission systems with polarization-dependent loss (PDL), which results from optical elements used in repeating nodes in long-haul transmission, such as reconfigurable-optical-add-drop-multiplexers [5], and affects the system performance [6–8]. The transfer function of a PDL element is not unitary, as shown in the following section. Therefore, when PDM signal lights pass through a PDL element, the output polarization states become nonorthogonal to each other, irrespective of whether the input states are orthogonal. Then, nonorthogonal PDM signals propagate through a transmission fiber. It is not certain whether the Manokov equation can describe the behavior of such nonorthogonal PDM signal lights.
Recently, the authors presented a nonlinear wave equation, modified from the Manokov equation, for wavelength/polarization multiplexed signals with random polarization states for each wavelength [9]. Following this work, the present study extends the wave equation to fiber transmission systems with PDL wherein the polarization states of PDM signals are nonorthogonal. A formula of the nonlinear polarization, that includes the degree of nonorthogonality caused by the PDL, is derived. Calculations using the derived formula are also preformed, the result of which suggests that the nonlinear interaction between non-orthogonal PDM signals is scarcely different from that between orthogonal PDM signals, thus, theoretically confirming that the Manakov equation or modified Manakov equation can be applied to nonorthogonal PDM signal lights.
2. Nonorthogonality due to polarization-dependent loss
Priory to consider nonlinear interaction between nonorthogonally polarized PDM signal lights, we review how the nonorthogonality results from PDL.
In the Jones vector space, the transfer matrix of an optical element with PDL is expressed as
(1)$${\mathbf T} = \left( {\begin{array}{{c}} 1\\ {\sqrt T } \end{array}} \right), $$
where T (< 1) is the transmittance of the y component relative to the x component. We assume that orthogonally polarization-multiplexed signal lights, that is expressed as Jones vectors as below, pass through the above optical element: (2a)$${{\mathbf E}_1}(\textrm{in}) = {E_1}\left( {\begin{array}{{c}} {\cos \phi }\\ {{e^{i\Delta }}\sin \phi } \end{array}} \right), $$
(2b)$${{\mathbf E}_2}(\textrm{in}) = {E_2}\left( {\begin{array}{{c}} { - {e^{ - i\Delta }}\sin \phi }\\ {\cos \phi } \end{array}} \right), $$
where E1 and E2 are the amplitude of the two signals, and ϕ and Δ indicate the signal polarization states. Using Eqs. (1) and (2), the output signal lights are expressed as (3a)$${{\mathbf E}_1}(\textrm{out}) = {\mathbf T}{{\mathbf E}_1}(\textrm{in}) = {E_1}\left( {\begin{array}{{c}} {\cos \phi }\\ {\sqrt T {e^{i\Delta }}\sin \phi } \end{array}} \right), $$
(3b)$${{\mathbf E}_2}(\textrm{out}) = {\mathbf T}{{\mathbf E}_2}(\textrm{in}) = {E_2}\left( {\begin{array}{{c}} { - {e^{ - i\Delta }}\sin \phi }\\ {\sqrt T \cos \phi } \end{array}} \right), $$
whose inner product is (4)$${{\mathbf E}_1}{(\textrm{out})^\ast } \cdot {{\mathbf E}_2}(\textrm{out}) ={-} {E_1}^\ast {E_2}{e^{ - i\Delta }}\cos \phi \sin \phi (1 - T) \ne 0. $$
The above consideration indicates that orthogonally polarization-multiplexed signals become nonorthogonal through a PDL element.The unit vectors representing the polarization states of the output light, given by Eq. (3), are expressed as
(5a)$${{\mathbf e}_1} = {A_1}\left( {\begin{array}{{c}} {\cos \phi }\\ {\sqrt T {e^{i\Delta }}\sin \phi } \end{array}} \right), $$
(5b)$${{\mathbf e}_2} = {A_2}\left( {\begin{array}{{c}} { - {e^{ - i\Delta }}\sin \phi }\\ {\sqrt T \cos \phi } \end{array}} \right), $$
where A1 and A2 are the normalization factors expressed as (6a)$${A_1} = \frac{1}{{\sqrt {{{\cos }^2}\phi + T{{\sin }^2}\phi } }}, $$
(6b)$${A_2} = \frac{1}{{\sqrt {T{{\cos }^2}\phi + {{\sin }^2}\phi } }}. $$
These unit vectors can be expressed by the unit vectors along the x and y axes ex and ey as
(7a)$${{\mathbf e}_1} = {A_1}\cos \phi \cdot {{\mathbf e}_x} + {A_1}\sqrt T {e^{i\Delta }}\sin \phi \cdot {{\mathbf e}_y}, $$
(7b)$${{\mathbf e}_2} ={-} {A_2}{e^{ - i\Delta }}\cos \phi \cdot {{\mathbf e}_x} + {A_2}\sqrt T \sin \phi \cdot {{\mathbf e}_y}. $$
Using the above unit vectors, we formalize the nonlinear polarization induced by nonorthogonal PDM signal lights in the following sections.3. Wavelength/polarization multiplexed signals
We consider nonlinear interactions in wavelength/polarization division multiplexing (WDM/PDM) fiber transmission. In order to treat WDM signals as well as PDM signals, we introduce the unit vectors of the signal polarization states of the kth wavelength light as follows:
(8a)$${{\mathbf e}_{k1}} = {A_{k1}}\cos {\phi _k} \cdot {{\mathbf e}_x} + {A_{k1}}\sqrt {{T_k}} {e^{i{\Delta _k}}}\sin {\phi _k} \cdot {{\mathbf e}_y}, $$
(8b)$${{\mathbf e}_{k2}} ={-} {A_{k2}}{e^{ - i{\Delta _k}}}\sin {\phi _k} \cdot {{\mathbf e}_x} + {A_{k2}}\sqrt {{T_k}} \cos {\phi _k} \cdot {{\mathbf e}_y}, $$
with (9a)$${A_{k1}} = \frac{1}{{\sqrt {{{\cos }^2}{\phi _k} + {T_k}{{\sin }^2}{\phi _k}} }}, $$
(9b)$${A_{k2}} = \frac{1}{{\sqrt {{T_k}{{\cos }^2}{\phi _k} + {{\sin }^2}{\phi _k}} }}, $$
where Tk represents PDL for the kth wavelength light, and 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.Using the above unit vectors, the amplitude vector at the kth wavelength is expressed as
(10)$$\scalebox{0.86}{$\displaystyle\begin{aligned} {{\mathbf E}_k} &= E_x^{(k)}{{\mathbf e}_x} + E_y^{(k)}{{\mathbf e}_y} = {E_{k1}}{{\mathbf e}_{k1}} + {E_{k2}}{{\mathbf e}_{k2}}\\ &= {E_{k1}}({A_{k1}}\cos {\phi _k} \cdot {{\mathbf e}_x} + {A_{k1}}\sqrt {{T_k}} {e^{i{\Delta _k}}}\sin {\phi _k} \cdot {{\mathbf e}_y}) + {E_{k2}}( - {A_{k2}}{e^{ - i{\Delta _k}}}\sin {\phi _k} \cdot {{\mathbf e}_x} + {A_{k2}}\sqrt {{T_k}} \cos {\phi _k} \cdot {{\mathbf e}_y})\\ &= ({E_{k1}}{A_{k1}}\cos {\phi _k} - {E_ - }{A_{k2}}{e^{ - i{\Delta _k}}}\sin {\phi _k}){{\mathbf e}_x} + ({E_{k1}}{A_{k1}}{e^{i{\Delta _k}}}\sin {\phi _k} + {E_{k2}}{A_{k2}}\cos {\phi _k}){{\mathbf e}_y}, \end{aligned}$}$$
where Ek1 and Ek2 are the amplitudes in the signal polarization states, respectively, and Ex(k) and Ey(k) are those along the x and y axes, respectively, at the kth wavelength. From Eq. (10), {Ek1, Ek2} and {Ex(k), Ey(k)} are related as follows: (11a)$$E_x^{(k)} = {E_{k1}}{A_{k1}}\cos {\phi _k} - {E_{k2}}{A_{k2}}{e^{ - i{\Delta _k}}}\sin {\phi _k}, $$
(11b)$$E_y^{(k)} = {E_{k1}}{A_{k1}}{e^{i{\Delta _k}}}\sin {\phi _k} + {E_{k2}}{A_{k2}}\cos {\phi _k}, $$
and subsequently, (12a)$${E_{k1}} = E_x^{(k)}\frac{1}{{{A_{k1}}}}\cos {\phi _k} + E_y^{(k)}\frac{1}{{{A_{k1}}\sqrt {{T_k}} }}{e^{ - i{\Delta _k}}}\sin {\phi _k}, $$
(12b)$${E_{k2}} ={-} E_x^{(k)}\frac{1}{{{A_{k2}}}}{e^{i{\Delta _k}}}\sin {\phi _k} + E_y^{(k)}\frac{1}{{{A_{k2}}\sqrt {{T_k}} }}\cos {\phi _k}. $$
Using the above notations, the x and y components of the wavelength multiplexed lights are expressed as
(13a)$${E_x} = \sum\limits_k {E_x^{(k)}{e^{ik2\pi \Delta ft}}} = \sum\limits_k {({E_{k1}}{A_{k1}}\cos {\phi _k} - {E_{k2}}{A_{k2}}{e^{ - i{\Delta _k}}}\sin {\phi _k}){e^{ik2\pi \Delta ft}}}, $$
(13b)$${E_y} = \sum\limits_k {E_y^{(k)}{e^{ik2\pi \Delta ft}}} = \sum\limits_k {({E_{k1}}{A_{k1}}\sqrt {{T_k}} {e^{i{\Delta _k}}}\sin {\phi _k} + {E_{k2}}{A_{k2}}\sqrt {{T_k}} \cos {\phi _k}){e^{ik2\pi \Delta ft}}}, $$
and their intensities are expressed as (14a)$$\scalebox{0.86}{$\displaystyle\begin{aligned} |{E_x}{|^2} &= {\left|{\sum\limits_k {({E_{k1}}{A_{k1}}\cos {\phi_k} - {E_{k2}}{A_{k2}}{e^{ - i{\Delta _k}}}\sin {\phi_k}){e^{ik2\pi \Delta ft}}} } \right|^2} = \sum\limits_k {|{E_{k1}}{A_{k1}}\cos {\phi _k} - {E_{k2}}{A_{k2}}{e^{ - i{\Delta _k}}}\sin {\phi _k}{|^2}} \\ &= \sum\limits_k {\{ |{E_{k1}}{|^2}A_{k1}^2{{\cos }^2}{\phi _k} + |{E_{k2}}{|^2}A_{k2}^2{{\sin }^2}{\phi _k} - ({E_{k1}}E_{k2}^\ast {e^{i{\Delta _k}}} + E_{k1}^\ast {E_{k2}}{e^{ - i{\Delta _k}}}){A_{k1}}{A_{k2}}\cos {\phi _k}\sin {\phi _k}\} } ,\end{aligned}$}$$
(14b)$$\begin{aligned} |{E_y}{|^2} &= \sum\limits_k {\{ |{E_{k1}}{|^2}A_{k1}^2{T_k}{{\sin }^2}{\phi _k} + |{E_{k2}}{|^2}A_{k2}^2{T_k}{{\cos }^2}{\phi _k}} \\ &+ ({E_{k1}}E_{k2}^\ast {e^{i{\Delta _k}}} + E_{k1}^\ast {E_{k2}}{e^{ - i{\Delta _k}}}){A_{k1}}{A_{k2}}{T_k}\cos {\phi _k}\sin {\phi _k}\} ,\end{aligned}$$
where Δf is the frequency spacing between WDM channels. In expanding the above equations, the beat frequency components, such as (k – k’)Δf for k ≠ k’, are neglected, considering the phase-mismatch in normal-dispersion fibers [9].4. Nonlinear polarization
Using the above notations, a formula of the nonlinear polarization in the signal polarization state at the sth wavelength can be obtained. First, the nonlinear polarization along the x and y axes at the sth wavelength in a birefringent fiber is given by [10, Appendix A]
(15a)$$P_{s(x)}^{\textrm{NL}} = \gamma (|{E_x}{|^2} + |{E_y}{|^2})E_x^{(s)} - \frac{\gamma }{3}|{E_y}{|^2}E_x^{(s)}, $$
(15b)$$P_{s(y)}^{\textrm{NL}} = \gamma (|{E_x}{|^2} + |{E_y}{|^2})E_y^{(s)} - \frac{\gamma }{3}|{E_x}{|^2}E_y^{(s)}, $$
where γ is the nonlinear constant standardly used for fiber nonlinearity. Substituting Eqs. (11) and (14), Eq. (15) is rewritten as (16a)$$\begin{aligned} P_{s(y)}^{\textrm{NL}} &= \gamma ({E_{k1}}{e^{i{{\Delta }_k}}}{A_{k1}}\sqrt {{T_k}} \sin {\phi _k} + {E_{k2}}{A_{k2}}\sqrt {{T_k}} \cos {\phi _k})\\ &\times \sum\limits_k {\left\{ {|{E_{k1}}{|^2} + |{E_{k2}}{|^2} - {A_{k1}}{A_{k2}}(1 - {T_k})\cos {\phi _k}\sin {\phi _k}({E_{k1}}E_{k2}^*{e^{i{{\Delta }_k}}} + E_{k1}^*{E_{k2}}{e^{ - i{{\Delta }_k}}})} \right\}} \\ &- \frac{\gamma }{3}({E_{k1}}{e^{i{{\Delta }_k}}}{A_{k1}}\sqrt {{T_k}} \sin {\phi _k} + {E_{k2}}{A_{k2}}\sqrt {{T_k}} \cos {\phi _k})\\ &\times \sum\limits_k {\left\{ {|{E_{k1}}{|^2}A_{k1}^2{{\cos }^2}{\phi _k} + |{E_{k2}}{|^2}A_{k2}^2{{\sin }^2}{\phi _k}} \right.} \\ &\left. { - {A_{k1}}{A_{k2}}\cos {\phi _k}\sin {\phi _k}({E_{k1}}E_{k2}^*{e^{i{{\Delta }_k}}} + E_{k1}^*{E_{k2}}{e^{ - i{{\Delta }_k}}})} \right\}, \end{aligned}$$
(16b)$$\begin{aligned} P_{s(y)}^{\textrm{NL}} &= \gamma ({E_{k1}}{e^{i{\Delta _k}}}{A_{k1}}\sqrt {{T_k}} \sin {\phi _k} + {E_{k2}}{A_{k2}}\sqrt {{T_k}} \cos {\phi _k})\\ &\times \sum\limits_k {\{{|{E_{k1}}{|^2} + |{E_{k2}}{|^2} - {A_{k1}}{A_{k2}}(1 - {T_k})\cos {\phi_k}\sin {\phi_k}({E_{k1}}E_{k2}^\ast {e^{i{\Delta _k}}} + E_{k1}^\ast {E_{k2}}{e^{ - i{\Delta _k}}})} \}} \\ &- \frac{\gamma }{3}({E_{k1}}{e^{i{\Delta _k}}}{A_{k1}}\sqrt {{T_k}} \sin {\phi _k} + {E_{k2}}{A_{k2}}\sqrt {{T_k}} \cos {\phi _k})\\ &\times \sum\limits_k {\{{|{E_{k1}}{|^2}A_{k1}^2{{\cos }^2}{\phi_k} + |{E_{k2}}{|^2}A_{k2}^2{{\sin }^2}{\phi_k}} } \\& { - {A_{k1}}{A_{k2}}\cos {\phi_k}\sin {\phi_k}({E_{k1}}E_{k2}^\ast {e^{i{\Delta _k}}} + E_{k1}^\ast {E_{k2}}{e^{ - i{\Delta _k}}})} \}.\end{aligned} $$
Similarly to Eq. (12), the nonlinear polarizations in the signal polarization states es1 and es2 are related to those along the x and y axes as follows:
(17a)$$P_{s1}^{\textrm{NL}} = P_{s(x)}^{\textrm{NL}}\frac{1}{{{A_{s1}}}}\cos {\phi _s} + P_{s(y)}^{\textrm{NL}}\frac{1}{{{A_{s1}}\sqrt {{T_s}} }}{e^{ - i{\Delta _s}}}\sin {\phi _s}, $$
(17b)$$P_{s2}^{\textrm{NL}} ={-} P_{s(x)}^{\textrm{NL}}\frac{1}{{{A_{s2}}}}{e^{i{\Delta _s}}}\sin {\phi _s} + P_{s(y)}^{\textrm{NL}}\frac{1}{{{A_{s2}}\sqrt {{T_s}} }}\cos {\phi _s}, $$
where Ps1NL and Ps2NL are the nonlinear polarizations in es1 and es2 at the sth wavelength, respectively. Substituting Eq. (16), Eq. (17a) is rewritten as (18)$$\begin{aligned} P_{s1}^{\textrm{NL}} &= \gamma (|{E_{s1}}{|^2} + |{E_{s2}}{|^2}){E_{s1}} - \frac{1}{2}\gamma (E_{s1}^2E_{s2}^\ast {e^{i{\Delta _s}}} + |{E_{s1}}{|^2}{E_{s2}}{e^{ - i{\Delta _s}}}){A_{s1}}{A_{s2}}(1 - {T_s})\sin 2{\phi _s}\\ &+ \gamma {E_{s1}}\sum\limits_{k \ne s} {\left\{ {|{E_{k1}}{|^2} + |{E_{k2}}{|^2} - \frac{1}{2}{A_{k1}}{A_{k2}}(1 - {T_k})\sin 2{\phi_k}({E_{k1}}E_{k2}^\ast {e^{i{\Delta _k}}} + E_{k1}^\ast {E_{k2}}{e^{ - i{\Delta _k}}})} \right\}} \\ &- \frac{\gamma }{3}\left[ {\frac{1}{4}|{E_{s1}}{|^2}{E_{s1}}A_{s1}^2(1 + {T_s}){{\sin }^2}2{\phi_s} + |{E_{s2}}{|^2}{E_{s1}}A_{s2}^2({T_s}{{\cos }^4}{\phi_s} + {{\sin }^4}{\phi_s})} \right.\\ &\quad\quad + \frac{1}{2}{A_{s1}}{A_{s2}}({T_s}{\cos ^2}{\phi _s} - {\sin ^2}{\phi _s})\sin 2{\phi _s}(E_{s1}^2E_{s2}^\ast {e^{i{\Delta _s}}} + |{E_{s1}}{|^2}{E_{s2}}{e^{ - i{\Delta _s}}})\\ &\quad\quad + \frac{1}{2}|{E_{s1}}{|^2}{E_{s2}}{A_{s1}}{A_{s2}}{e^{ - i{\Delta _s}}}({\cos ^2}{\phi _s} - {T_s}{\sin ^2}{\phi _s})\sin 2{\phi _s}\\ &\quad\quad + \frac{1}{2}|{E_{s2}}{|^2}{E_{s2}}\frac{{A_{s2}^3}}{{{A_{s1}}}}{e^{ - i{\Delta _s}}}({\sin ^2}{\phi _s} - {T_s}{\cos ^2}{\phi _s})\sin 2{\phi _s}\\&\quad\quad - \frac{1}{4}A_{s2}^2(1 + {T_s}){\sin ^2}2{\phi _s}({E_{s1}}|{E_{s2}}{|^2} + E_{s1}^\ast E_{s2}^2{e^{ - i2{\Delta _s}}})\\& + ({E_{s1}}{\cos ^2}{\phi _s} - \frac{1}{2}{E_{s2}}\frac{{{A_{s2}}}}{{{A_{s1}}}}{e^{ - i{\Delta _s}}}\sin 2{\phi _s})\sum\limits_{k \ne s} {(|{E_{k1}}{|^2}A_{k1}^2{T_k}{{\sin }^2}{\phi _k} + |{E_{k2}}{|^2}A_{k2}^2{T_k}{{\cos }^2}{\phi _k})} \\&+ ({E_{s1}}{\sin ^2}{\phi _s} + \frac{1}{2}{E_{s2}}\frac{{{A_{s2}}}}{{{A_{s1}}}}{e^{ - i{\Delta _s}}}\sin 2{\phi _s})\sum\limits_{k \ne s} {(|{E_{k1}}{|^2}{A_{k1}}^2{{\cos }^2}{\phi _k} + |{E_{k2}}{|^2}{A_{k2}}^2{{\sin }^2}{\phi _k})} \\ &\left. { + \frac{1}{2}({E_{s1}}\cos 2{\phi_s} - {E_{s2}}\frac{{{A_{s2}}}}{{{A_{s1}}}}{e^{ - i{\Delta _s}}}\sin 2{\phi_s})\sum\limits_{k \ne s} {{A_{k1}}{A_{k2}}{T_k}\sin 2{\phi_k}({E_{k1}}E_{k2}^\ast {e^{i{\Delta _k}}} + E_{k1}^\ast {E_{k2}}{e^{ - i{\Delta _k}}})} } \right]. \end{aligned}$$
This equation represents the locally induced nonlinear polarization in the es1 polarization state at the sth wavelength, which is a function of ϕk and Δk representing the signal polarization states. Along the transmission fiber, the polarization states randomly vary owing to slight birefringence. Therefore, we average the above expression with respect of ϕk and Δk on the Poincaré sphere [9], the result of which is
(19a)$$\begin{aligned} P_{s1}^{\textrm{NL}} &= \gamma {E_{s1}}|{E_{s1}}{|^2}\left\{ {1 - \frac{1}{3}(1 + {T_s})\left\langle {\frac{{{{\cos }^2}{\phi_s}{{\sin }^2}{\phi_s}}}{{{{\cos }^2}{\phi_s} + {T_s}{{\sin }^2}{\phi_s}}}} \right\rangle } \right\}\\ &+ \gamma |{E_{s2}}{|^2}{E_{s1}}\left\{ {1 - \frac{1}{3}\left\langle {\frac{{{T_s}{{\cos }^4}{\phi_s} + {{\sin }^4}{\phi_s}}}{{{T_s}{{\cos }^2}{\phi_s} + {{\sin }^2}{\phi_s}}}} \right\rangle + \frac{1}{3}(1 + {T_s})\left\langle {\frac{{{{\cos }^2}{\phi_s}{{\sin }^2}{\phi_s}}}{{{T_s}{{\cos }^2}{\phi_s} + {{\sin }^2}{\phi_s}}}} \right\rangle } \right\}\\ &+ \frac{5}{6}\gamma {E_{s1}}\sum\limits_{k \ne s} {(|{E_{k1}}{|^2} + |{E_{k2}}{|^2})} ,\end{aligned} $$
where < > denotes averaging with respect to ϕs on the Poincaré sphere. It is noted in this equation that Tk of k ≠ s disappears as a result of the averaging. Similarly, the nonlinear polarization in the signal polarization state es2 after the averaging is derived as (19b)$$\begin{aligned} P_{s2}^{\textrm{NL}} &= \gamma {E_{s2}}|{E_{s1}}{|^2}\left\{ {1 - \frac{1}{3}\left\langle {\frac{{{{\cos }^4}{\phi_s} + {T_s}{{\sin }^4}{\phi_s}}}{{{{\cos }^2}{\phi_s} + {T_s}{{\sin }^2}{\phi_s}}}} \right\rangle + \frac{1}{3}(1 + {T_s})\left\langle {\frac{{{{\cos }^2}{\phi_s}{{\sin }^2}{\phi_s}}}{{{{\cos }^2}{\phi_s} + {T_s}{{\sin }^2}{\phi_s}}}} \right\rangle } \right\}\\& + \gamma |{E_{s2}}{|^2}{E_{s2}}\left\{ {1 - \frac{1}{3}(1 + {T_s})\left\langle {\frac{{{{\cos }^2}{\phi_s}{{\sin }^2}{\phi_s}}}{{{T_s}{{\cos }^2}{\phi_s} + {{\sin }^2}{\phi_s}}}} \right\rangle } \right\}\\& + \frac{5}{6}\gamma {E_{s1}}\sum\limits_{k \ne s} {(|{E_{k1}}{|^2} + |{E_{k2}}{|^2})} .\end{aligned} $$
In case of no PDL (i.e., Ts = 1), these expressions become
(20a)$$P_{s1}^{\textrm{NL}} = \frac{8}{9}\gamma \left\{ {|{E_{s1}}{|^2} + |{E_{s2}}{|^2} + \frac{{15}}{{16}}\sum\limits_{k \ne s} {(|{E_{k1}}{|^2} + |{E_{k2}}{|^2})} } \right\}{E_{s1}}, $$
(20b)$$P_{s2}^{\textrm{NL}} = \frac{8}{9}\gamma \left\{ {|{E_{s1}}{|^2} + |{E_{s2}}{|^2} + \frac{{15}}{{16}}\sum\limits_{k \ne s} {(|{E_{k1}}{|^2} + |{E_{k2}}{|^2})} } \right\}{E_{s2}}, $$
which are coincident with the expressions for WDM/PDM signals with orthogonal polarization states [9].In Eq. (19), there are four terms averaged over the polarization state. Although they appear to be somewhat different, some of them are identical as a result of the averaging, as follows [Appendix B]:
(21a)$$\left\langle {\frac{{{{\cos }^2}{\phi_s}{{\sin }^2}{\phi_s}}}{{{{\cos }^2}{\phi_s} + {T_s}{{\sin }^2}{\phi_s}}}} \right\rangle = \left\langle {\frac{{{{\cos }^2}{\phi_s}{{\sin }^2}{\phi_s}}}{{{T_s}{{\cos }^2}{\phi_s} + {{\sin }^2}{\phi_s}}}} \right\rangle, $$
(21b)$$\left\langle {\frac{{{{\cos }^4}{\phi_s} + {T_s}{{\sin }^4}{\phi_s}}}{{{{\cos }^2}{\phi_s} + {T_s}{{\sin }^2}{\phi_s}}}} \right\rangle = \left\langle {\frac{{{T_s}{{\cos }^4}{\phi_s} + {{\sin }^4}{\phi_s}}}{{{T_s}{{\cos }^2}{\phi_s} + {{\sin }^2}{\phi_s}}}} \right\rangle. $$
Subsequently, Eq. (19) is rewritten as
(22a)$$P_{s1}^{\textrm{NL}} = \gamma \left\{ {\left( {1 - \frac{1}{3}{p_1}} \right)|{E_{s1}}{|^2} + \left( {1 + \frac{1}{3}{p_1} - \frac{1}{3}{p_2}} \right)|{E_{s2}}{|^2} + \frac{5}{6}\sum\limits_{k \ne s} {(|{E_{k1}}{|^2} + |{E_{k2}}{|^2})} } \right\}{E_{s1}}, $$
(22b)$$P_{s2}^{\textrm{NL}} = \gamma \left\{ {\left( {1 + \frac{1}{3}{p_1} - \frac{1}{3}{p_2}} \right)|{E_{s1}}{|^2} + \left( {1 - \frac{1}{3}{p_1}} \right)|{E_{s2}}{|^2} + \frac{5}{6}\sum\limits_{k \ne s} {(|{E_{k1}}{|^2} + |{E_{k2}}{|^2})} } \right\}{E_{s2}}, $$
where the following parameters are introduced for simplifying the expressions: (23a)$${p_1} = (1 + {T_s})\left\langle {\frac{{{{\cos }^2}{\phi_s}{{\sin }^2}{\phi_s}}}{{{{\cos }^2}{\phi_s} + {T_s}{{\sin }^2}{\phi_s}}}} \right\rangle, $$
(23b)$${p_2} = \left\langle {\frac{{{T_s}{{\cos }^4}{\phi_s} + {{\sin }^4}{\phi_s}}}{{{T_s}{{\cos }^2}{\phi_s} + {{\sin }^2}{\phi_s}}}} \right\rangle. $$
Applying Eq. (22) to a conventional wave equation, the following nonlinear wave equations can be obtained:
(24a)$$\begin{aligned} \frac{{\partial {E_{s1}}}}{{\partial z}} &+ \frac{\alpha }{2}{E_{s1}} + i\frac{{\beta _{s1}^{(2)}}}{2}\frac{{{\partial ^2}{E_{s1}}}}{{\partial {t^2}}}\\& = i\gamma \left\{ {\left( {1 - \frac{1}{3}{p_1}} \right)|{E_{s1}}{|^2} + \left( {1 + \frac{1}{3}{p_1} - \frac{1}{3}{p_2}} \right)|{E_{s2}}{|^2} + \frac{5}{6}\sum\limits_{k \ne s} {(|{E_{k1}}{|^2} + |{E_{k2}}{|^2})} } \right\}{E_{s1}} ,\end{aligned} $$
(24b)$$\begin{aligned} \frac{{\partial {E_{s2}}}}{{\partial z}} &+ \frac{\alpha }{2}{E_{s2}} + i\frac{{\beta _{s2}^{(2)}}}{2}\frac{{{\partial ^2}{E_{s2}}}}{{\partial {t^2}}}\\ &= i\gamma \left\{ {\left( {1 + \frac{1}{3}{p_1} - \frac{1}{3}{p_2}} \right)|{E_{s1}}{|^2} + \left( {1 - \frac{1}{3}{p_1}} \right)|{E_{s2}}{|^2} + \frac{5}{6}\sum\limits_{k \ne s} {(|{E_{k1}}{|^2} + |{E_{k2}}{|^2})} } \right\}{E_{s2}}, \end{aligned} $$
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 nonlinear interaction between WDM/PDM signals with the nonorthogonality resulting from PDL is described by the above coupled equations.5. Calculation
In order to see the dependency of the nonlinear interaction on the nonorthogonality caused by PDL, we calculated the coefficients related to the PDL in the nonlinear polarization, that is, p1 and p2 given by Eq. (23), as a function of T. Figure 1 shows the results. The coefficients barely deviate from the values for orthogonally polarized signals, that is, 1/3 for p1 and 2/3 for p2, although they depend on T in the formulas. This calculation result suggests that the nonlinear interaction between nonorthogonal polarization-multiplexed signals is almost the same as that between orthogonally multiplexed ones. This may be owing to the averaging with respect to the polarization state, through which some of the terms including PDL are canceled out from Eqs. (18) to (19).
6. Summary
This study presented a formula describing nonlinear interaction between nonorthogonal polarization-multiplexed signal lights in fiber transmission with PDL. An expression of the nonlinear polarization was derived, which includes the degree of the nonorthogonality caused by PDL. Calculation indicating the effect of the nonorthogonality on the nonlinear polarization was also carried out, the result of which showed that the nonorthogonality or PDL scarcely changes the nonlinear interaction in effect. Therefore, this study theoretically confirmed that the Manakov equation or modified Manakov equation can be applied to nonorthogonal PDM signal lights in practice.
Appendix A
In this appendix, we review where Eq. (15) originates from.
In an isotropic material, the nonlinear polarization induced along the x axis from z-propagating lights is expressed as [11]
(25)$$P_x^{\textrm{NL}} = 3{c_{xxxx}}{E_x}{E_x}E_x^\ast{+} 3{c_{xxyy}}{E_x}{E_y}E_y^\ast{+} 3{c_{xyxy}}{E_y}{E_x}E_y^\ast{+} 3{c_{xyyx}}{E_y}{E_y}E_x^\ast , $$
where
cxijk (
i,
j,
k, = {
x,
y}) is a tensor component of the third-order nonlinear susceptibility and
Ex,y is the light amplitude along the
x or
y axis perpendicular to the propagation direction
z. The tensor components satisfy
cxxyy =
cxyxy =
cxyyx =
cxxxx/3 in an isotropic material. The above equation represents the locally induced nonlinear polarization, which then generates
x-polarized light. The
x-polarized nonlinear light generated at
z0, for example, is superimposed onto nonlinear lights that are generated at previous positions of
z <
z0 and propagate to
z0 with the propagation constant for
x-polarized light,
βx. Here, the propagation constant of a nonlinear polarization component 3
cxijkEiEjEk* is
βi +
βj –
βk. Therefore, when
βx =
βi +
βj –
βk (i.e., the phase matching condition in nonlinear optics), the locally generated nonlinear lights are superimposed in phase. In the nonlinear polarization expressed in Eq. (
25), the first three components satisfy the phase matching condition while the last one does not in a birefringent medium with
βx ≠
βy.
The phase mismatch effect is indicated by Δβ ≡ |βx – βy|. Under the condition of ΔβL > 2π where L is the medium length, nonlinear lights generated from a phase-mismatched nonlinear polarization component are phase-randomly superimposed, and are canceled out in total. In a nonlinear medium with a birefringence of Δn = 10−7, Lc = 2π/Δβ = 15 [m]. For such a medium with a length of $\ge {L_\textrm{c}}$, the last component in Eq. (25) can be neglected, and the nonlinear polarization is rewritten as
(26)$$\begin{aligned} P_x^{\textrm{NL}} &= 3{c_{xxxx}}{E_x}{E_x}E_x^\ast{+} 3{c_{xxyy}}{E_x}{E_y}E_y^\ast{+} 3{c_{xyxy}}{E_y}{E_x}E_y^\ast \\ &= 3{c_{xxxx}}|{E_x}{|^2}{E_x} + {c_{xxxx}}|{E_y}{|^2}{E_x} + {c_{xxxx}}|{E_y}{|^2}{E_x}\\ &= 3{c_{xxxx}}(|{E_x}{|^2} + |{E_y}{|^2}){E_x} - {c_{xxxx}}|{E_y}{|^2}{E_x}. \end{aligned} $$
This expression can be applied to a fiber segment with a length of $\ge {L_\textrm{c}}$. In nonlinear fiber optics, the nonlinear parameter γ is standardly used for indicating the nonlinear efficiency of a waveguide medium, which includes the mode confinement effect. Using this γ, Eq. (26) is rewritten as
(27)$$P_x^{\textrm{NL}} = \gamma (|{E_x}{|^2} + |{E_y}{|^2}){E_x} - \frac{\gamma }{3}|{E_y}{|^2}{E_x}, $$
which is Eq. (
15a). Equation (
15b) is similarly obtained.
Appendix B
This appendix shows the equivalency in the coefficients related to the polarization state, that is indicated in Eq. (21). For a stochastic variable dependent on the polarization state as ξ = f(ϕ = ψ/2) in general, its average for the polarization state uniformly distributed on the Poincaré sphere is given by [9]
(28)$$< \xi > = \frac{1}{2}\int\limits_0^\pi {f(\psi )\sin \psi \cdot d\psi }. $$
Using this formula, the left-hand side of Eq. (21a) is expanded as
(29a)$$\begin{aligned} \left\langle {\frac{{{{\cos }^2}\phi {{\sin }^2}\phi }}{{{{\cos }^2}\phi + T{{\sin }^2}\phi }}} \right\rangle &= \frac{1}{2}\left\langle {\frac{{1 - {{\cos }^2}2\phi }}{{1 + T + (1 - T)\cos 2\phi }}} \right\rangle \\& = \frac{1}{4}\int\limits_0^\pi {\frac{{1 - {{\cos }^2}\chi }}{{1 + T + (1 - T)\cos \chi }} \cdot \sin \chi d\chi } = \frac{1}{4}\int\limits_{ - 1}^1 {\frac{{1 - {x^2}}}{{1 + T + (1 - T)x}}dx} \\& = \frac{1}{4}\int\limits_0^1 {\frac{{1 - {x^2}}}{{1 + T - (1 - T)x}}dx} + \frac{1}{4}\int\limits_0^1 {\frac{{1 - {x^2}}}{{1 + T + (1 - T)x}}dx}. \end{aligned}$$
On the other hand, the right-hand side is expanded as
(29b)$$\begin{aligned} \left\langle {\frac{{{{\cos }^2}\phi {{\sin }^2}\phi }}{{T{{\cos }^2}\phi + {{\sin }^2}\phi }}} \right\rangle &= \frac{1}{2}\left\langle {\frac{{1 - {{\cos }^2}2\phi }}{{1 + T - (1 - T)\cos 2\phi }}} \right\rangle \\& = \frac{1}{4}\int\limits_0^\pi {\frac{{1 - {{\cos }^2}\chi }}{{1 + T - (1 - T)\cos \chi }} \cdot \sin \chi d\chi } = \frac{1}{4}\int\limits_{ - 1}^1 {\frac{{1 - {x^2}}}{{1 + T - (1 - T)x}}dx} \\ &= \frac{1}{4}\int\limits_0^1 {\frac{{1 + {x^2}}}{{1 + T + (1 - T)x}} \cdot dx} + \frac{1}{4}\int\limits_0^1 {\frac{{1 - {x^2}}}{{1 + T - (1 - T)x}} \cdot dx}. \end{aligned} $$
Comparing the above equations, Eq. (
21a) is confirmed.
Regarding Eq. (21b), its left- and right-hand sides are expanded as follows:
(30a)$$\scalebox{0.86}{$\displaystyle\begin{aligned} \left\langle {\frac{{T{{\cos }^4}\phi + {{\sin }^4}\phi }}{{T{{\cos }^2}\phi + {{\sin }^2}\phi }}} \right\rangle &= \frac{1}{4}\int\limits_{ - 1}^1 {\frac{{1 + T - 2(1 - T)x + (1 + T){x^2}}}{{1 + T + (1 - T)x}}dx} \\ &= \frac{1}{4}\int\limits_0^1 {\frac{{1 + T - 2(1 - T)x + {{(1 + T)}^2}}}{{1 + T + (1 - T)x}}dx} + \frac{1}{4}\int\limits_0^1 {\frac{{1 + T - 2(1 - T)x + {{(1 + T)}^2}}}{{1 + T - (1 - T)x}}dx} ,\end{aligned}$}$$
(30b)$$\scalebox{0.86}{$\displaystyle\begin{aligned}\left\langle {\frac{{{{\cos }^4}\theta + T{{\sin }^4}\theta }}{{{{\cos }^2}\theta + T{{\sin }^2}\theta }}} \right\rangle &= \frac{1}{4}\int\limits_{ - 1}^1 {\frac{{1 + T + 2(1 - T)x + (1 + T){x^2}}}{{1 + T - (1 - T)x}}dx} \\ &= \frac{1}{4}\int\limits_0^1 {\frac{{1 + T + 2(1 - T)x + (1 + T){x^2}}}{{1 + T - (1 - T)x}}dx} + \frac{1}{4}\int\limits_0^1 {\frac{{1 + T + 2(1 - T)x + (1 + T){x^2}}}{{1 + T + (1 - T)x}}dx},\end{aligned}$}$$
Subsequently, Eq. (
21b) is confirmed.
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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