Open access peer-reviewed chapter

Non-Manakovian Propagation in Optical Fiber

Written By

Lothar Moeller

Submitted: 24 November 2021 Reviewed: 14 February 2022 Published: 06 July 2022

DOI: 10.5772/intechopen.103694

From the Edited Volume

The Nonlinear Schrödinger Equation

Edited by Nalan Antar and İlkay Bakırtaş

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Abstract

Solving the nonlinear Schrödinger equation or similar calculus is essential for designing today’s long-haul optical communication systems. Associated numerical and analytical approaches were extensively studied over the past four decades; simplifications and adaptions for various applications and purposes have been introduced. Optical fibers installed in long-haul systems possess nonideal features such as birefringence, which some of these adaptions partially address to improve the simulation accuracy. But as the fiber birefringence frequently and randomly changes along a link, finding a mathematical solution is a more complex problem and beyond the task of dealing with the nonlinear nature of the corresponding equations. Recently, a novel propagation phenomenon related to the polarization evolution of a signal has been observed. In links with considerable length, i.e., bridging transatlantic or transpacific distances, the polarization state of a light wave is impacted by neighboring signals via the Kerr nonlinearity in glass. Established formalisms for describing polarization effects based on the nonlinear Schrödinger equation cannot fully capture this phenomenon. Here we discuss a first-order calculus for this problem. We start with high-level reviews of experimental observations to introduce the phenomenon and ways to model regular nonlinear propagation. Then we present a first-order calculus to describe the statistics behind the phenomenon by specifically discussing the interplay between fiber birefringence and fiber nonlinearities.

Keywords

• nonlinear depolarization
• Kerr nonlinearity
• polarization
• single-mode fiber
• Manakov equation
• Manakov-PMD equation
• coupled NL Schrödinger equations

1. Introduction

Long-distance optical telecommunications using standard single-mode fiber (SSMF) is economically [1] one of the most important industrial applications, which requires an accurate solution to the nonlinear (NL) Schrödinger equations for developing competitive products. Modern fiber communications, the basis of all backbone networks, enable global long-reach and high-capacity data exchange such as the WWW and state-of-the-art systems provide 10s of Tb/s capacity per fiber over transpacific distances without electrical signal regeneration [2, 3].

By simulating the NL Schrödinger equation or similar approaches, the industry assesses potential commercial link design options for network operations [4, 5]. The chosen optical power levels balance the generated NL signal distortions with sufficiently high optical signal-to-noise ratios (OSNR) to guarantee certain bit error rates (BERs) on the receive side.

Throughout the half-century-long history of SSMF, various mathematical approaches for describing NL signal propagation have been proposed where all of them are somewhat related to the (coupled) NL Schrödinger equation(s) [6, 7]. These models vary in terms of complexity, their intended purposes, and user friendliness. For example, a more refined analysis of light propagation in SSMF for telecom application requires a polarization dependent field representation. Today’s coherent signaling technique, which exploits two orthogonal polarization states at same optical frequency to encode data information and the unavoidable birefringence of SSMF, requires a vector field representation of the light mode. But simultaneously considering both aspects, the polarization of light and its random changes along a regular link due to fiber birefringence lead to an extended set of equations that are for most applications impractical to solve exactly.

The high end of the “manageable” equation sets, the “Manakov-PMD equation” addresses NL propagation in a fiber with randomly varying birefringence [8]. For the current generation of communication systems, it provides quite accurate results for expected NL signal distortions. But recently, a fiber phenomenon, to which we refer to as nonlinear depolarization (NLDP) of light in fiber, has been observed and does not conform to the Manakov-polarization mode dispersion (PMD) equation. Here we discuss some relevant experimental aspects of this phenomenon and its impact on the mathematical description of NL propagation in realistic, i.e., birefringent fiber. The Manakov-PMD equation is in some ways an advanced form of the NL Schrödinger equation as it additionally considers PMD effects [9, 10], and on the other hand, a simplification as it uses averaged quantities over distance to describe the randomly changing fiber birefringence.

In this chapter, we describe an algorithm for solving the NL Schrödinger equation in vector form when the field variables are randomly and rapidly alternated by the stationary and linear properties (birefringence) of the channel. In other words, some coefficients of the coupled NL Schrödinger equation (CNLS) become distance dependent and describe the changing features of the glass medium along the propagation path.

We start with a brief overview on commonly used modeling for NL signal propagation (Manakovian propagation) in SSMF, report on a high-level view of experimental results that do not conform to such formalism (non-Manakovian propagation) and discuss an algorithm that yields some analytic quantities for a theoretical description of the later. Although not fully technically correct, we use the SSMF terminology in our chapter [11] even when we also mean other fiber types such as large effective area fiber (LEAF [12]) that are often installed in long-haul communication systems.

Our focus is to develop a mathematical formalism for NLDP that yields quantities that are observable in typical industrial test beds. Certainly, more sophisticated experimental setups can be built to characterize other features of NLDP. We report the experimental conditions to a degree that produces a qualitative understanding of the phenomenon and will reference further details in the literature.

2. Nonlinear propagation equations

Back in 1972 and far before commercial applications of optical communications became relevant, Sergey Manakov1 suggested that a careful consideration of NL pulse propagation is required for accurate signal modeling in fiber. He proposed a set of coupled differential equations that to a large degree can capture the impact of the fiber’s Kerr nonlinearity on a signal’s evolution [13]. A slowly varying envelope constituted by the two orthogonal variables Ax and Ay represents the pulse in space and time domains:

Axz+β1Axt+jβ222Axt2+α2Ax=Ax2+Ay2AxE1
Ayz+β1Ayt+jβ222Ayt2+α2Ay=Ax2+Ay2AyE2

where z and t are the propagation direction and time, respectively. The optical features of the fiber are characterized by a,β1,β2,andγ, which refer to the fiber attenuation, propagation constant, group velocity, and Kerr nonlinearity [14], respectively. By multiplying Ax and Ay with the optical carrier, which typically resides around 194 THz (1.5 μm wavelength), the optical field can be obtained.

In the field of telecom the above listed formulas are called the Manakov Equation (ME). Literature uses this name for both its scalar version and a two-dimensional version for polarization representation of a signal. The ME was heuristically found in the sense that the left side, known from linear transmission theory, has been extended by a source term on its right side describing the Kerr nonlinearity. It assumes that the nonlinearities are weak and proportional to the signal’s intensity.

The ME has been highly successfully applied to NL signal propagation and can explain phenomena such as optical solitons and nonlinear polarization rotation (NLPR [15, 16]) that were subjects of intensive research until about the end of the 1990 [17, 18, 19, 20]. But the ME in the form outlined above is restricted in twofold aspects compared with a more accurate propagation modeling that modern telecom applications require:

(a) It assumes a rotational symmetry of the fiber, i.e., without birefringence. But commercial SSMFs, even those with the lowest amounts of birefringence, demand for modeling that adapts the ME’s left side (the linear propagation features) to simulate typical evolutions of a signal’s state of polarization (SOP). We address this issue by substituting β1 and β2 with polarization-dependent constants β1x,β1y,β2x and β2y and

(b) other than the ME hypothesizes, the fiber Kerr NL of glass is significantly polarization dependent. The impact of an orthogonal polarization component needs to be weighted by a factor of 2/3.

Both adjustments in the ME lead to a simplified version of the so-called coupled NL Schrödinger equations (CNLSs):

Axz+β1xAxt+jβ222Axt2+α2Ax=Ax2+23Ay2AxE3
Ayz+β1yAyt+jβ222Ayt2+α2Ay=Ay2+23Ax2AyE4

The extended version differs by terms on the right side, which model fast oscillations of the nonlinearity (not shown). In our further considerations, those terms can be neglected; and the polarization dependence of β2 is small and can be ignored. Strictly speaking, the CNLSs hold only for a single piece of fiber with linear birefringence, referred to in the literature as waveplate. Different velocities of light in both optical axes (principal states) of a waveplate lead in general to SOP changes for a cw tone passing through a single plate and complicate the calculus.

A model of a realistic fiber link would typically involve a concatenation of many waveplates, each with their optical axes randomly oriented. Solving the CNLS becomes for such a scenario a cumbersome process. On the other hand, it had been observed that NL differential equations of Manakovian type (Eqs. (1) and (2)) sufficiently capture experimental results such as soliton propagation. This led to the assumption that the ME can be derived from the CNLS when its polarization-dependent NL source term is averaged in a way that it simulates polarization scrambling caused by fiber birefringence [21, 22]. Several derivations have been published, which yield an ME with an 8/9 reduced Kerr nonlinearity as fitting parameter:

Axz+β1Axt+jβ222Axt2+α2Ax=89Ax2+Ay2AxE5
Ayz+β1Ayt+jβ222Ayt2+α2Ay=89Ay2+Ax2AyE6

Note, the fiber birefringence does not appear anymore in the form of polarization-dependent propagation constants, and the x, y-indexes should not be identified with the coordinates of the lab frame. Nevertheless, advanced effects such as NL polarization rotation of two cw tones can be quantitatively described by such equations.

The heuristically determined ME with and without reduced Kerr nonlinearity has become an integral part of soliton theory and is often used in today’s system modeling due to its relatively simple form and highly accurate results. Based on this success, the published attempts for deriving the ME from the more fundamental CNLSs have gained wide acceptance [21]. Here, however, we challenge their formalism as they rely on an elusively compelling mathematical argument and show that the solution spaces of the ME with reduced Kerr nonlinearity and of the section-wise solved CNLS for long birefringent fibers diverge from each other. This difference matters in case of NLDP.

The Manakov-PMD equation developed toward the end of the 1990 is cumbersome but allows to separate the principal effects by terms such as the Kerr nonlinearity, chromatic dispersion, nonlinear PMD, and linear PMD that contribute to the signal’s evolution. Its nonlinear PMD term can be ignored in applications for today’s regular telecom fibers, thus the remaining NL source term of the Manakov-PMD equation simplifies to the same form as known from the ME (Eqs. (5) and (6)). It differs from the latter by an additional term that accounts for linear PMD contributions. Nonlinear PMD refers to a situation where the fiber birefringence does not sufficiently scramble the SOP of a cw signal when it linearly propagates over distances that are comparable to the NL propagation length of the path. In our following considerations, we can (to good approximation) assume that a signal’s SOP gets strongly scrambled represented by a homogeneous coverage on the Poincare sphere while it propagates just a few kilometers. We can therefore neglect NL PMD.

Nevertheless, due to the same method of averaging that had been applied when deriving from the CNLS the ME with an 8/9 reduced Kerr nonlinearity, the Manakov-PMD equation does not represent NLDP fully correctly, regardless if it includes NL PMD or not.

3. Comparative polarimetry for detecting NLDP

The undersea communication industry continuously aims for a more precise modeling of signal propagation in fiber to enable longer unrepeated system spans and higher transport capacities, hence more cost-effective solutions. However, a noticeable discrepancy in channel capacity between measurable performance and predictions from the most advanced simulation tools remains, which attracts significant research in propagation modeling mostly focusing on pure optical effects [23] but lately also considering weak acousto-optic interactions [24, 25]. Here we discuss NL depolarization of light (NLDP), a recently observed non-Manakovian transmission phenomenon as a potential candidate to narrow down this discrepancy [26]. Unpolarized optical noise rapidly changes via the fiber’s Kerr nonlinearity the SOP from a fully polarized cw light by inducing antisymmetric phase noise in both of its orthogonal polarization states. These fluctuations become resolvable with a new generation of high-speed polarimeters [27] and do not average out over wide noise bandwidths but grow with propagation distance.

Prior to applying the CNLS to NLDP, we introduce the phenomenon by providing some experimental evidence that will prompt the discussion of quantities needed for developing our algorithm. Back in 2017, two measurements were performed on an undersea cable that connects Brazil and Florida to demonstrate NLDP (Figure 1a) on a real communication link [26, 28]. A cw light launched from Brazil and looped back in Florida propagates via the opposite fiber link to its origin. There are about 200 optical amplifiers in each direction to compensate for fiber attenuation but cause weak NL propagation in the first few tens of kilometer fiber length directly after each repeater.

Undersea cables on the ocean floor undergo environmental impact such as motion due to water currents, seismic vibrations, and temperature fluctuations, etc. These environmental factors lead to small SOP motions on the receive side, trackable with a low-speed polarimeter, and represented as velocities in histograms. We define the amount of the time derivative from the corresponding normalized Stokes vector as “SOP speed.” Typical SOP speeds range on the order of a few rad/s (Figure 1b). In this experiment, a cw light (ECL, External Cavity Laser) also referred to as probe (or signal) is launched together with unpolarized Amplified Spontaneous Emission (ASE, loading) into the cable input. The ASE spectrum covers the whole repeater bandwidth (∼4.5 THz) except for a narrow central gap with ∼100 GHz width where the probe resides (Figure 1c). Without loading, the probe would pick up almost the entire repeater output power resulting in strong NL propagation, mainly self phase modulation (SPM) [29], which can produce unstable SOPs. With loading, the probe contributes less than 1% of the total repeater output power and NL effects such as Brillouin scattering [30] are avoided. The electrical detection bandwidth of the low-speed polarimeter (a few 10s of Hz) averages out all fast SOP motions; while increasing its bandwidth (a few 10s of MHz) produces much wider SOP speed histograms under identical experimental conditions (Figure 1d).

It would be incorrect to attribute the entire width of this histogram solely to NL interactions between the loading and the probe, as any noise during the detection corrupts the polarimeter, leading to artifacts in SOP speeds. In our experiment, signal-ASE beat noise is the main contributor and [31] biases the SOP detection. Every repeater adds small amounts of ASE that raise the noise floor within the spectral gap. This ASE then mixes with the probe on the four photodiodes of the polarimeter. Even a hypothetical constant SOP of the probe on the receive side will appear in a histogram with nonzero width due to the omnipresent signal-ASE beat noise. We refer to this noise-induced SOP speed (NISS) as an artifact since improving the optical signal-to-noise ratio (OSNR) at the polarimeter input would reduce the width of its histogram. However, probe power constraints, inevitable added repeater noise during transmission, and practical limitations on tighter filtering yield OSNR levels, which result in artificially broadened SOP speed histograms that partially obscure NLDP.

Instead, we demonstrate NLDP-induced fast SOP changes by means of a comparison technique. We contrast the SOP speed histograms from the transmitted probe with one from a reference signal that possesses equal power, equal ASE, and equal OSNR but bypasses the undersea cable (reference, btb). This reference is obtained by superimposing the transmitter signal with the noise output of the cable in Brazil and launched btb into the receiver (ECL switched to reference path, Figure 1a). Within the resolution capabilities (∼2 GHz) of our optical spectrum analyzer (OSA), both the transmitted probe and the reference spectra are identical (Figure 1d). However, the SOP speed histogram for the probe is significantly wider compared with the histogram for the reference (Figure 1e) as NL interactions fluctuated the probe’s SOP. Even when the optical noise floor in the btb experiment is subject to added ASE (∼5 dB, Figure 1d), the corresponding histogram does not expand to the same width of the probe’s plot (Figure 1e).

It is reasonable to assume a scaling of the NLDP magnitude with the probe’s transmission length and the loading power, as both determine the strength of the Kerr nonlinearity. Such parametric NLDP studies become feasible with a lab test bed based on a recirculating fiber loop (RFL, Figure 2a) [32]. In RFLs, a fast optical switch allows to launch a signal into a fiber link whose output is coupled back into its input. After a programmable number of round trips inside the loop, the signal is released via the switch and analyzed.

By programming the loop’s timing gate, SOP speed histograms are determined (Figure 2c) for propagation lengths of approx. 1023 km, 3069 km, 5115 km, 10,230 km, and 20,460 km further referred to as 1 Mm, 3 Mm, …, 20 Mm transmissions and correspond to 1, 3, 5, 10, and 20 loop circulations, respectively. For longer transmissions, the corresponding btb measurements show wider NISS histograms as the received OSNRs decay. The probe’s SOP speed histograms broaden with propagation length due to NISS and NLDP. We hypothesize both as statistically independent processes and visualize NLDP by subtracting the reference’s NISS from the probe’s SOP speed variance (Figure 2c, inset). This quantity monotonically increases, indicating a growing NLDP with transmission distance. A linear fit reasonably resembles the measured NLDP variances as a function of the propagation length. Each recording consists of about 100 MSamples and was repeated multiple times to verify stable measurement readings. We found experimentally for the short-term reproducibility of all measured SOP speed variances a relative error smaller than 1%.

Without NL interaction with the loading, the probe would linearly propagate. Altering the nominal probe power of ∼−5.2 dBm at the repeater output by ±3 dB has shown an insignificant dependence of σ2NLDP on SPM-based effects at 10 Mm transmission distance. Hence, Brillouin scattering or modulation instability (known to be the weakest NL fiber process) can be ruled out as origin for NLDP.

As discussed in Section 4.7, NLDP is formed on long-range nonlinear optical interactions that become observable in the spectral domain. Distortions mainly generated within the first section of a transmission fiber connected to a repeater output substantially interfere among each other when propagating along a multi span link. Qualitatively spoken, the spectral features of the probe’s Stokes vector depend on the transmission distance—more precisely, the further the signal propagates, the leaner the Stokes vector spectrum becomes. In our study, we define as the probe’s spectrum the Fourier transform of the autocorrelation of its normalized, three-dimensional Stokes vector.

In our experimental verification, we utilize the same RFL as in the aforementioned setup. But instead of using a polarimeter, a polarization scrambling interferometer connected to a photodetector followed by an RF spectrum analyzer is deployed (Figure 3a). It can be shown that averaging many RF spectra yields the Stokes vector spectrum [35]. For emulated transmission distances between 1 and 20 Mm spectral width (FWHM), between 16.1 and 3.3 MHz, respectively, were observed. This demonstrates the spectral contraction over propagation length (Figure 3b).

An analogy to Fabry-Perot etalons [36] whose filter width narrows with the number of interfering rays inside its cavity maybe helpful for understanding the spectral shaping of the Stokes vector spectrum with transmission distance. While a higher reflectivity of the etalon’s mirrors produces more interfering rays, more link spans lead to more coherently superimposing distortions in the receiver plane that cause spectral compression.

4. Applying the coupled NL Schrödinger equations to NLDP

Techniques for solving the CNLS for a short piece of fiber with constant linear birefringence have been extensively discussed in the literature [37]. SSMF can be envisioned as a concatenation of many fiber pieces with linear birefringence. The length of each piece and the orientation of its optical axes follow known statistics. Our goal is to calculate closed-form solutions for quantities that describe the underlying statistics of the reported NLDP effects based on the aforementioned fiber models. For the sake of simplicity, we take advantage of certain experimental conditions that justify a significantly reduced formalism. The conceptional simplifications relate to NL propagation, the repeater functionality, and the fiber model.

We assume NL signal propagation but consider its impact as relatively weak, which effectively addresses the operational range of today’s telecom systems. This allows us to model NL distortions as first-order perturbations that propagate linearly through the path once they have been generated.

Every repeater adds small amounts of ASE to the loading. While this extra noise does not significantly change the spectral shape of the loading, it slightly impacts its temporal correlation features, which will be ignored.

We will partially diverge from the picture of discrete and concatenated waveplates that form a fiber. For SSMF, it is appropriate to imagine smooth transitions between the single waveplates. However, we will consider a discrete waveplate model to discuss local SOP rotations and apply the theory of PMD statistics to cope with long-distance SOP correlations. These assumptions should not impact our main conclusions.

4.1 Sorting the Kerr nonlinearity in even and odd operators

For simplicity, we consider a weak cw field axy (probe) residing in a narrow spectral gap of a surrounding, fully unpolarized and co-propagating strong ASE field Axy (loading) with a boxcar-shaped spectrum (Figure 4a). As stated, significantly less than 1% of the total signal power stems from the probe. The known CNLS [38] for propagation in z direction within a waveplate, given here in complete form, can approximate this scenario well:

AxΣz+β1xAxΣt+jβ222AxΣt2+α2AxΣ=AxΣ2+23AyΣ2AxΣ+jγ13AxΣAyΣ2ej2ΔβzE7
AyΣz+β1yAyΣt+jβ222AyΣt2+α2AyΣ=AyΣ2+23AxΣ2AyΣ+jγ13AyΣAxΣ2e+j2ΔβzE8
withAxyΣ=Axy+axy;axyAxy,E9

where the wavelength-independent α, β1x(y)−1, β2, and γ denote the attenuation, polarization-dependent group velocities, dispersion coefficient, and the Kerr nonlinearity of a waveplate, respectively. As previously mentioned, the high modal birefringence Δβ of regular SMF induces fast oscillating of the terms at the far-right side, which leads to ineffective NL interference. This has been discussed in great detail for the derivation of the Manakov-PMD equation [8]; however, it is negligible in our analysis and thus left out in the following.

We assume that the loading modulates the probe, but the probe has no impact on the loading. In the limit of a negligibly small probe power, such interaction can be justified and simplifies the Kerr nonlinearity. Two sets of equation pairs follow, for the two cases axy=0 and 1axy0. Subtracting both sets, neglecting second-order terms, and considering coupling conditions yield an equation set that describes the motion of the probe induced by the loading

axz+β1xaxt+jβ2x22axt2+α2ax=jγ2Ax2+23Ay2axE10
ayz+β1yayt+jβ2y22ayt2+α2ay=jγ2Ay2+23Ax2ay.E11

The remaining stochastic perturbation on the right side is decomposed into a symmetric and an antisymmetric term (Eqs. (12) and (13)) with respect to the loading’s field

axz+β1xaxt+jβ2x22axt2+α2ax=jγ43Ax2+Ay2ax+23Ax2Ay2axE12
ayz+β1yayt+jβ2y22ayt2+α2ay=jγ43Ay2+Ax2ay+23Ay2Ax2ay.E13

While both describe weak and independently treatable NL interactions in a first-order perturbation calculus, the polarization-dependent sign of the antisymmetric perturbation (right sides of Eqs. (12) and (13) root-causes opposite phase noises in both principal axes that manifests experimentally as NLDP. Identical phase changes in both polarizations as produced by the symmetric perturbation do not alter the probe’s SOP but lead to NL phase noise. In the following, we discuss solutions for the pair

axz+β1axt+jβ222axt2+α2ax=jγ23Ax2Ay2axE14
ayz+β1ayt+jβ222ayt2+α2ay=jγ23Ay2Ax2ay.E15

The impact of birefringence has been disregarded in Eqs. (14) and (15) by replacing β1x(y) with a polarization-independent group velocity β1−1 but will be revisited at a later stage. That is, Eqs. (14) and (15) captures nonlinear SOP changes along a waveplate but does not include the much larger SOP changes caused by linear birefringence. The linear SOP changes are static and do not contribute to the measured SOP speeds. In practical applications, the right side represents a very fast fluctuating term as the noise components Axand Ay stem from stochastically independent processes with bandwidths in the range of several THz. Since we assume unpolarized loading, the quantity Axy2Ayx2 is on average zero. For our algorithm, we represent it on an evenly spaced frequency grid, which yields experimentally observable quantities to describe some statistical features of the probe axy.

4.2 Asymmetric phase noise in first-order approximation

The right side of Eqs. (14) and (15) weakly perturbs the probe by adding a first-order correction term a1x(y)(z,t) to it. We write the perturbation by means of the undistorted fields of the probe ax(y)(z) and the loading Ax(y)(z,t). The latter’s components with amplitudes Ax(y) form a comb Eqs. (16) and (17) on an evenly spaced grid with an infinitesimally small angular frequency pitch ω (Figure 4a). For a single span system, the probe ax(y)(z,t) in first-order development and the ASE field read

A0xzt=m=NNAxmejkmzmωteα2zE16
A0yzt=m=NNAymejkmzmωteα2zE17
axzta0x+a1xzteα2z=a0x+la1xlzejlωteα2zE18
ayzta0y+a1yzteα2z=a0y+la1ylzejlωteα2zE19
withAxym=0form<Nu,m>N,z0andA0xzt=axzt=0forz<0E20

where km=β1+β222 stands for the propagation constant of a component at ‘’. We synthesize the Kerr nonlinearity in Eqs. (14) and (15) as a sum to address the impact of low-frequency beat noises among its terms. Due to phase matching conditions, this noise alone can efficiently interact with the probe and is used to redefine the perturbation term in Eqs. (14) and (15) as

Ax2Ay2m,lAxl,meαzejβ1+β2lmω2zlωtE21
Ay2Ax2m,lAyl,meαzejβ1+β2lmω2zlωtE22
withAxyl,m=Axym+l2Axyml2Ayxm+l2Ayxml2,lNu.E23

Here stands for the angular frequency spacing between the two beating ASE tones at ωm±l2 and the probe, which is typically in the THz range. Due to coupling inefficiency, any beating among the two noise tones can be neglected when its frequency resides beyond a few tens of MHz. Eqs. (21) and (22) hold separately for every l and m, and its solution for a single span system can be written by means of a Green’s function

a1xyl,mzt=j23γ0zejklzziαziAxyl,mejβ1+β2lmω2zilωta0xydziE24

According to this Ansatz, the NL distortions are generated as a kind of “wave packet” in fiber sections of incremental length dzi, propagate thereafter linearly through the span, and coherently superimpose in the receiver plane. We portion out the integral of Eq. (24) into a sum of infinitesimal short waveplates dzi (Eq. (25)) to analyze NL interactions in the presence of Polarization Mode Dispersion (PMD), which originates from fiber birefringence. At the span output at L0 holds

δ1xyl,mL0t=a1xyl,mL0ta0xy=j23γejβ1L0lωtiAxym,leαziejβ2lmω2ziej12β2l2ω2L0zidzi.E25

Alternating the sign ofl conjugates its right side, except for its last and typically negligible small exponent (a1xyl,mL0ta1xyl,mL0t). Hence, pairing contributions at ±l results into a correction with a 90 phase offset relative to the undistorted probe. Therefore, all pairs of NL distortions stemming from single waveplates generate pure phase oscillations in the receiver plane εL0t=δ1xyl,mL0t+δ1xyl,mL0t at an angular frequency . As Axym,l=Ayxm,l holds, the oscillations in both orthogonal polarizations are 180 out of phase and cause SOP fluctuations. We define for later purposes a temporal autocorrelation as

φi,kl,mτ=δ1x,il,mL0t+τδ1x,kl,mL0t+δ1y,il,mL0t+τδ1y,kl,mL0t,E26

where the indices i,k denote different waveplates and denotes the averaging over time and fields, which involves reestablishing the birefringent fiber features in our model as detailed below.

4.3 Correlations of asymmetric phase noise in birefringent fiber

For deriving correlations between phase noises generated at different propagation distances of the probe, we focus on a single span system and extend the results to a multiple span link. Modeling a birefringent fiber as a concatenation of waveplates uses Eq. (25) to determine the correlation among incremental distortions stemming from two different plates. Our x(y)-coordinate system is congruent with the fast (slow) axis of a waveplate, i.e., it rotates and follows the plates’ orientations along the propagation path. Our model incorporates birefringence, originating from, e.g., axis-specific group velocities (β1x ≠ β1y), by means of a Jones matrix that transforms the input SOPs from the probe and the noise components Axym when traversing a waveplate. A Jones matrix R¯i of a waveplate shall be given by a unitary matrix

R¯i=R11iR12iR12iR11iwithR11i2+R12i2=1.E27

Witha0xya0=a0xa0y and I¯=1001, an NL distortion generated in waveplate k with length dzk concisely reads at the span output

a1,km,lL0t=a1x,km,lL0ta1y,km,lL0t=j23γejβ1L0lωtejβ2lmω2zkdzk×k=iML0R¯iAx,km,lI¯eαzki=1k1R¯ia0,E28

where Ax,km,l und ML0stand for the undepleted noise components inside of waveplate kconstituted by Eq. (23), and the total number of waveplates, respectively. The correlation between contributions from two consecutive waveplates follows as

a1,k+1m,lL0a1,km,lL0=49γ2ejβ2lmω2zkzk+1eαzk+zk+1dzkdzk+1×Ax,k+1m,lI¯R¯ka0,kAx,km,lR¯kI¯a0,k,E29

where a0,k and represent the undistorted and undepleted probe field within waveplate k and the field-averaged scalar product, respectively. We restore PMD in our fiber model by using matrices R¯i,R¯i obeying Eq. (27) and transforming wavelength-dependent the probe and ASE fields:

Ax,k+1m,lI¯R¯ka0,kR¯kAx,km,lI¯a0,k=R11k2R12k2R11k2R12k2Hma0,k2E30
withHm=Axm"2Axm"2+Aym2Aym"2,E31

where double primes indicate statistically independent noises. Since ml holds, we can evaluate Eq. (30) just at m but must treat the original two noise components at frequencies m±l2ω as uncorrelated. The theory of PMD statistics [39] specifies the distance-dependent decorrelation from two SOPs of two co-propagating cw tones at different wavelengths, which provides a correlation between the matrix elements

R11k2R12k2R11k2R12k2=13e13ωm2τρ2L,E32

where mω, τρ, and L=zk+1zk are the angular frequency spacing between the probe and the two noise components, the mean fiber DGD per √length (Differential Group Delay), and the propagation distance, respectively. PMD effects across frequency intervals of size ∼lω such as the spectral width of the received and distorted probe or spacings between two efficiently beating noise components are negligibly small. Hence, the sum of all incremental phase oscillations correlates as defined by Eq. (26) at the span output like

φl,mτ=427γ2ejlωτHmi,kejβ2lmω2zizkαzi+zke13ωm2τρ2zizkdzidzk.E33

The above outlined calculus assumes two consecutive birefringent waveplates k,k+1. But it holds for any pair of further spaced waveplates, indexed i,k with ik+1, as well. Since a matrix product R¯=R¯iR¯k+2R¯k+1 of intermediately located waveplates can be expressed by a single unitary matrix that fulfills Eq. (27), the conclusion from Eq. (30) will equally hold and leads to Eq. (33).

4.4 Fiber PMD constitutes NLDP

System PMD imposes cutoff conditions via the Gaussian for the number of interacting waveplates addressed by the double sum of Eq. (33). Without this constrain, the sum tends to zero as its complex exponential function causes averaging for sufficiently small α. For a single span system with L0α1 (typically tens of kilometers) and relatively short waveplates (α1, typically tens of meters), we will replace the double sum with an integral.

Experimentally observed NLDP-caused SOP features such as scattering angles and speed as their time derivatives are detected after O/E conversion of the optical fields and conveniently reported in Stokes space (Section 3). To derive such quantities, we will confine the optical autocorrelation density by introducing electrical low-pass filtering, which represents the detection process, and then convert the result into Stokes space. In Jones space, the density of the optical autocorrelation (Eq. (33)) at reads for sufficiently long propagation distances L0

φl,mτ=427γ2Hmejlωτ2α+eα+13ωm2τρ2zejβ2lmω2zdz=427γ2Hmejlωταα+13ωm2τρ2α+13ωm2τρ22+β2lmω22E34

and its integration over m,l yields the autocorrelation for the total phase noise.

So far, we have considered a single span system, i.e., one optical amplifier boosts the probe and loading prior of their launch into an as infinite long assumed transmission fiber. Today’s undersea communication cables can consist of a few hundred spans to bridge transpacific distances of up to about 15 Mm length. In contrast to terrestrial systems, they are strictly modularly designed, which eases our modeling. To keep the calculation effort at an introductory level, we further assume a large enough fiber attenuationα, thus all NL propagation fades away far before the span end. Additionally, our amplification is assumed to be a noiseless process. For a cable with NS spans, the autocorrelation then reads

φNsl,mτ=427γ2ejlωτHm×p=0,q=0Ns,Nsi,kejβ2lmω2zizk+L0pqαzi+zke13ωm2τρ2zizk+L0pqdzidzkE35

with zi+L0p and zk+L0q as the positions of two interacting waveplates.

For approximating the interleaved summations, we take advantage of the assumed large fiber attenuation, which forms cutoff conditions for the zi and zk. When one of the two coordinates or both are large enough but still significantly smaller than L0, the corresponding summands do not substantially contribute to the overall sum. Thus, a replacement of the amount in the exponents by

zizk+L0pqzizk+L0pqE36

can be justified and yields

φNsl,mτ427γ2ejlωτHm×i,kejβ2lmω2zizkαzi+zke13ωm2τρ2zizkdzidzk×p=0,q=0Ns,Nsejβ2lmω2L0pqe13ωm2τρ2L0pqE37
=427γ2eilωταHm×α+13ωm2τρ2α+13ωm2τρ22+β2lmω22×Nsn=NsNsejβ2lmω2L0ne13ωm2τρ2L0nΛnNs,E38

where Λ. denotes the triangular function. For the degree of accuracy, we follow in our modeling, it is sufficient to approximate it with an exponential function of even area and substitute both sums by integrals to obtain at least trends that show how NLDP qualitatively depends on system parameters.

φNsl,mτ427γ2eilωταHmα+13ωm2τρ2α+13ωm2τρ22+β2lmω22×Nsn=ejβ2lmω2L0ne13ωm2τρ2L0+2Nsn.E39

Further we approximate the sum with a Fourier integral and find for the density of the optical correlation

φNsl,mτ827γ2HmNsejlωταL01α+13ωm2τρ2213ωm2τρ2+2NsL011+β2lmω2α+13ωm2τρ2211+β2lmω213ωm2τρ2+2NsL02.E40

4.5 Detecting optical fields in the electrical domain

For the sake of simplicity, we consider two extreme cases for the electrical detection bandwidth Ωe of our polarimeter. In one scenario its bandwidth tends to infinity and in the other, it strongly filters the photocurrents. In both casesτS1Ωe should hold for its sampling rate. In typical lab experiments, the electrical detection bandwidth is the only parameter that can be practically tuned over a larger range without the need for readjusting other model assumptions. For example, reducing the optical bandwidth of a system (∼4.5 THz) at launch or the launch power into a span usually distorts the assumed boxcar-shaped spectrum of the loading.

We model the low-pass characteristic of the polarimeter with a Lorentzian curve, thus the autocorrelations in the optical and electrical domain interrelate as

φNselecτ=11+Ωe2φNsl,mτdωldωm,E41

Provided sufficiently short sampling periods τS, compared with the inverse of the autocorrelation’s spectral width2, the exponential in Eq. (40)ejlωτ1jlωτ12lωτ2 can be approximated in τ. Its linear term vanishes after the integration (Eq. (41)) due to symmetry aspects. The second order in τS of φτSelec determines the variance of the stochastic SOP speed as shown below and reads

φNselec0φNselecτS=11+Ωe2φNsl,m0φNsl,mτsdωldωm,E42

which tends for the case of wide bandwidth detection (Ωe) and under the assumptions of typical system parameters as explained in Section 3 toward

φNselec0φNselecτSπ81γ2NsαL0β23τS2τρ2Prep21ΩmaxΩmin2lnΩmaxΩmin,E43

where Prep represents the total launch power of the loading at the fiber input.

For the case of detection at a reduced electrical bandwidth (Ωe13Ωminτρ2β2), we find

φNselec0φNselecτSπ27γ2NsαL0Ωe2Ωmax2τS2β2αPrep2ln3α21τρΩmin.E44

As Eq. (44) rapidly declines for small Ωe2, NLDP stays hidden in the study of environmentally driven SOP fluctuations in undersea cables (Figure 1b), which had been performed at a detection bandwidth below 100 Hz.

4.6 Transforming phase noise from Jones space to stokes space

So far, NLDP has been characterized in Jones space for which the CNLSs hold. But for convenience in general, quantities such as the SOP speed and the SOP scattering angle [40] are usually discussed and experimentally obtained in Stokes space. To transform the incremental field distortions, determined by means of the Jones calculus and the CNLS, into Stokes space, we represent the assumed normalized probe’s Jones vector anormNsL0t=axNsL0tayNsL0tT at the fiber output by

anormNsL0t=axNsL0tayNsL0t=cosϑcosθ+jsinϑsinθsinϑcosθjcosϑsinθE45

with ϑ=ϑ0+δϑt,2ϑ0<πand θ=θ0+δθt,4θ0<πwhere ϑ0,θ0 obey known distributions [41] to uniformly cover the Poincare sphere. For the SOP density Ψ holds

Ψ2ϑ02θ0d2ϑ0d2θ0=14πrect2ϑ02πrect2θ0πcos2θ0d2ϑ0d2θ0.E46

NLDP causes small temporal fluctuations δϑt,δθt whose autocorrelations must equal δϑt+τSδϑt=δθt+τSδθt=φJτS due to averaging caused by birefringence fluctuations along the long propagation path. The corresponding three-dimensional Stokes vector with unity lengthS0=1 to Eq. (45) reads

SNsL0t=ax2ay2+2Raxay2ImaxayNsL0t=cos2ϑcos2θsin2ϑcos2θsin2θNsL0tE47

and can be analyzed with respect to the impact of δϑt and δθt. One finds

St+τSSt2=403φJ0φJτS.E48

In Jones space we get

anormt+τSanormt2=4φJ0φJτS.E49

Thus, combing both results add up to

St+τSSt2=103anormt+τSanormt2E50

Identifying anormt+τSanormt2=2φNselec0φNselecτS yields the variance of the NLDP-induced SOP speed in Stokes space

St+τSSt2τS22π81γ2NsαL0β23τρ2Prep21ΩmaxΩmin2lnΩmaxΩmin,E51

which depends on the PMD of the link and differentiates NLDP from NL polarization rotation (NLPR) [15]. NLPR is a phenomenon between two cw tones where one impacts the SOP of the other. However, its fundamental equations do not include fiber features such as PMD and chromatic dispersion.

4.7 NLDP-induced stokes vector spectrum of the probe

In analogy to the relationship between temporal and spectral features of an electrical signal in time and frequency domain, we define the spectrum of the probe’s Stokes vector by the Fourier transform of its temporal autocorrelation

St+τSSt=1203φJ0+203φJτS1+203φJτSΦf.E52

For our following derivations, we revisit Eq. (52). It does not include several approximations we have meanwhile applied in our calculus and can therefore be considered as a more accurate starting point. But we simplify the integration over the loading’s optical spectrum by neglecting its central gap and assume infinite integral bounds, i.e., Hm=Hm¯=const holds

φNsl,mτ=427γ2eilωτα2Hm¯11+13αωm2τρ211+β2lmω2α+13ωm2τρ22×Nsn=NsNsejβ2lmω2L0ne13ωm2τρ2L0nΛnNs.E53

For typical system parameters, Eq. (53) is dominated by the term e13ωm2τρ2L0n, which sets even for low fiber PMD (∼50 fs/√km) and small |n| > 0 a strong cutoff criterion for the integration. In this case the first two fractions on the right side will be ignored and the integration yields

φNslτejlωτn=Ns,n0Ns1ne3β224τρ2ωl2L0nΛnNs.E54

A narrowing of the probe’s Stokes vector spectrum with an increasing number of system spans can be qualitatively understood by defining its spectral width as

1φNs00φNsl0dωllnNs1Ns1.E55

This quantity decreases for large enough and increasing Ns . NL distortions generated along the transmission link interfere, which leads to spectral shaping of the Stokes vector spectrum. Such kind of spectral compression has been experimentally observed (Section 3). The more spans a link consists of, the more nonlinear distortions in form of wave packets superpose in the receiver plane and steepen the pedestal of the averaged RF spectra (Figure 3b) [35].

5. Non-Manakovian transmission

We continue to assume that the CNLSs provide a sufficiently accurate basis for signal modeling in a single birefringent waveplate and will compare in the following their nonlinearity with corresponding terms of often-used simplifications such as the Manakov equation [13], the Manakov-PMD equation [8], and GN theory [23]. It turns out that the nonlinearity used in our NLDP calculus differs significantly from simplified forms often exploited in today’s system simulations. For this purpose, we examine a known transformation of the CNLS into the ME for regular birefringent fiber, which leads to an inconsistency in the foundation of nonlinear propagation theory. This commonly applied assumption biases microscopic NL polarization effects. As for non-Manakovian effects such as the NLDP, we like to refer to when propagating signals cannot be sufficiently accurately described by applying equations of Manakovian type or the Manakov-PMD equation, which possess averaged and symmetric Kerr nonlinearities.

5.1 Shortfalls of an averaged Kerr nonlinearity

Here we review key steps for introducing an averaged Kerr nonlinearity into the CNLS and point out the associated shortfalls. The two main concepts [22, 41] to derive an averaged Kerr nonlinearity can be found in the literature. We follow a well-documented scheme in [41], which considers a piece of isotropic fiber (with zero birefringence) and neglect for now linear propagation effects such as attenuation and all forms of dispersion. Furthermore, like all published derivations we assume for AxyΣ monochromatic fields at ω=0, which some papers paraphrase as fully polarized light3. In this case Eqs. (7) and (8) reduces to polarization-dependent differentials that equal small NL distortions

AxΣz=jγAxΣ2+23AyΣ2AxΣ+jγ13AxΣAyΣ2E56
AyΣz=jγAyΣ2+23AxΣ2AyΣ+jγ13AyΣAxΣ2.E57

For normalized variables

AxΣ2+AyΣ2=1E58

Eqs. (56) and (57) can concisely be written as

AxΣz=jγ2AxΣ+AxΣAxΣ2+AyΣ2E59
AyΣz=jγ2AyΣ+AyΣAxΣ2+AyΣ2.E60

Similar as in Section 4.6, we evaluate the magnitude of the polarization state-dependent NL distortion by expressing the field with a Jones vector

AxΣAyΣ=cosϑcosε+jsinϑsinεsinϑcosεjcosϑsinεe.E61

But here, the time-independent variables ϑ,ϵ,ψare functions of the distance z. As empirically observed, the SOP of a cw light propagating through randomly and rapidly changing fiber birefringence is uniformly distributed on the Poincare sphere with corresponding densities for the angles ϑ,ε as assumed for Eq. (46).

Inserting Eq. (61) into the left sides of Eqs. (59) and (60) leads to a differential term for the phase

AxΣAyΣzψE62

that can be evaluated as scalar when multiplying both sides of the equation by AxΣAyΣ. Averaging the differentials across the Poincare sphere by means of Eq. (46) must equate to zero for terms including zϑand zε due to symmetry considerations. Thus, on the left only a term in zψ remains. On the right, AxΣ2+AyΣ2 is factored out, which is no longer assumed to be limited to unity, but represents time- and distance-dependent power levels of a signal. When adding the time-dependent differentials on the left sides that describe linear propagation effects and including the absorption term, then the ME as stated in form of Eqs. (5) and (6) with an averaged Kerr NL follows. As this derivation assumes an isotropic fiber, β1,β2 must be defined as x-, y-independent. Some closed-form solutions of Eqs. (5) and (6) are known (solitons). Its nonlinearity is symmetric in the field components. But strictly speaking, the above derivation holds only for cw light and so Eqs. (5) and (6). The reintroduction of Eq. (58) as time- and distance-dependent quantity without expanding term (62) by additional corresponding differentials violates the derivation’s assumptions and complicates it. Literature justifies the above derivation with lacking phase matching of mixing terms that can be neglected. However, if the field contains components that are closely separated, a low-frequency beat tone can occur. When sufficient phase matching is present (as in the case of NLDP), measurable modulation of the probe can appear. Additionally, the derivation does not mirror correlations among differentials in zϑ or zε taken at two different positions separated by PMD. But as such, they are essential for understanding the spectral properties of NLDP.

When choosing Eqs. (5) and (6) as starting point for the splitting of the overall NL operator as explicated by Eqs. (12) and (13), the symmetric and asymmetric NL perturbation terms change their relative weights. Hence, both phenomena NLDP and NL phase noise cannot be represented at the same time using the same effective average Kerr nonlinearity.

5.2 Manakovian simulators in telecom

A common technique in modern optical communications is to polarization multiplex two orthogonal channels at same wavelength, which maximizes the spectral efficiency of a system. When representing in a simplified picture transmitted data symbols per channel by optical wave packets, their instantaneous common receive SOP is equally blurred by NLDP in both azimuth and polar angles on the Poincare sphere. Especially, fast SOP motions in azimuthal direction (assuming the individual channels possess polarizations aligned with the x-y coordinates) impair coherent cross talk at high receive OSNR for advanced modulation formats and reduce established limits for the channel capacity. Current research on capacity limits of fiber channels has not explicitly considered NLDP. These theories apply Shannon’s theorem while computing NL signal distortions by means of the Manakov equation [42, 43, 44, 45, 46, 47, 48, 49]. They need to be revisited when more accurate estimates are required.

To ease computations, most of the industrial link simulations resort to a type of ME when estimating NL transmission penalties. For signals with low-density constellation (e.g., PM-QPSK), the small NLDP-induced SOP scattering has little performance impact and a Manakovian simulation can be a good approximation. As reference, for a link with moderate Kerr nonlinearity and transpacific transmission distance, we experimentally found an apex angle ∼11° for NLDP-induced scattering on the Poincare sphere [40].

The Manakov-PMD equation as defined in [8] includes the 8/9 factor for an averaged Kerr nonlinearity. For regular birefringent fiber, its NL PMD term becomes irrelevant and only its linear PMD term distinguishes it from an ordinary ME. However, the averaging of its Kerr nonlinearity that yields its 8/9 weighting questions its claim of universal acceptance as a governing model.

The GN model, another widely discussed approach for simulating optical communication channels, is a technique to analytically solve the ME for a weakly NL WDM system with D+ propagation [23, 50]. Its main advantage resides in the derivation of variances for NL field distortions that can be linearly added to noise powers (repeater ASE) in the SNR formula to determine a BER. Note, these NL field distortions should not be confused with polarization state noise generated by NLDP. The GN model does not include PMD effects, which lead to a linear SOP decorrelation between the probe and the beating ASE components [39]; but this decorrelation is essential for the foundation of the probe’s NLDP-induced Stokes vector spectrum, which is interferometrically formed by long-range NL interactions (Section 3 [35]).

6. Conclusions

Precise modeling of NL signal propagation in optical fibers is critical for maximizing the data capacity of long-haul communication systems. It balances signal powers and received OSNR to mitigate nonlinearities. Over the past five decades, simplified techniques have been developed to efficiently compute NL propagation in fiber. They adapt models for ideal or piece-wise linear birefringent fiber to simulate propagation paths with randomly varying birefringence.

Recently, a novel transmission phenomenon in fiber to which we refer to as NL DePolarization (NLDP) has been introduced. Unpolarized ASE depolarizes a co-propagating probe in long-haul communication systems and lab test beds due to the fiber Kerr nonlinearity. This phenomenon has proven elusive to simpler propagation modeling.

We have described NLDP by means of propagation-dependent SOP speed histograms. And under some simplifying assumptions, our outlined analytical model yields a closed-form solution for NLDP-induced SOP speed in single and multiple span systems. Although small compared with other polarization effects, this phenomenon leads to a qualitatively different microscopic understanding of nonlinear light propagation in fiber. An antisymmetric perturbation operator in the CNLS generates phase noises that produce the SOP fluctuations. A major aspect of our model forms the PMD dependence of NLDP, which fundamentally differentiates it from other NL polarization phenomena such as NL polarization rotation. NLDP is based on long-range NL interactions where contributions from Kerr nonlinearities interfere over long transmission distances. Counterintuitively, the NL-generated Stokes vector spectrum of a signal’s polarization narrows with increasing propagation length.

Our derivations show that in the case of NLDP (non-Manakovian propagation), the solution spaces of the CNLS and the Manakov equation do not converge as suggested by earlier work. Under consideration of NLDP, reassessing fiber channel capacity simulations that are utilizing Manakovian-type equations can be beneficial for scientific purposes and could show small performance offsets.

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Notes

• Sergey Valentinovich Manakov, Russian mathematician, *1948–†2012.
• With the current generation of commerically avaialbe high-speed polarimeters sampling periods ∼10 ns can be achieved while spectral widths of the autocorrelation typically reside in the few MHz range.
• Note, in the given derivation [22] on p. 29 it should read shortly above Eq. (3) “converts s3 σ3U to 1/3 s0 U.”

Written By

Lothar Moeller

Submitted: 24 November 2021 Reviewed: 14 February 2022 Published: 06 July 2022