Optical Fiber (FIBER) - INTERCONNECT Element

Optical fiber

Keywords

optical, bidirectional

Ports

Properties

General Properties

Standard Properties

Waveguide/Mode 1 Properties

Waveguide/Mode 2 Properties

Waveguide/Attenuation Properties

Waveguide/Dispersion Properties

Waveguide/Nonlinearities Properties

Waveguide/Nonlinearities/Raman Scattering Properties

Waveguide/Birefringence Properties

Numerical Properties

Numerical/Digital Filter Properties

Diagnostic Properties

Results

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Implementation Details

Overview

The propagation of light in a single mode fiber is governed by the following pair of coupled generalized nonlinear Schrödinger equations (CGNLSE) given in equation (1) [1,2]:

where:

A_x and A_y are the complex amplitudes of the signal in the slowly varying envelope approximation of the  optical modes polarized in the x and y directions, respectively.

x and  y are the transverse spatial coordinates

z is the longitudinal spatial coordinate

T is the retarded time given by

where t is the time coordinate, and v_g is the average group velocity of the polarized fiber modes.

and where v_gx and v_gy are the group velocities of the modes in polarized along the x and y directions.

ω₀ = 2πf₀ is the central angular frequency of the simulation band,

α is the absorption coefficient,

and β₂ and β₃ are given by:

Optical fiber manufacturers commonly provide values for dispersion, D,  and dispersion slope, D', in their data sheets, and β₂ and β₃ are related to them by the following relations:

where Δβ'_PMD is 0 if PMD effects are turned off and its calculation is described by equation (27) in the Polarization Mode Dispersion (PMD) section below.

and where the nonlinear parameter,  γ, is given by:

The effective area, A_eff, and nonlinear index, n₂ (ω₀ ), are usually provided by optical fiber manufacturers in their data sheets. The effective area can also be calculated from the transverse profile of the optical mode in the fiber, F(x,y) as follows:

The Raman time constant, T_R, can be calculated from:

where h_R (t) is the Raman impulse response, which is related to the Raman gain spectrum.

If the Raman gain spectrum is approximated by a Lorentzian then h_R (t) will take the form:

and the integral in equation (13) can be performed analytically.

Propagation Algorithm

In INTERCONNECT, the calculation is performed in block mode by discretizing and integrating equations (1a) and (1b) using the Symmetrized Split-Step Fourier Method (SSSFM) [1].

The input is comprised of a pair of input sequences of evenly spaced time samples of fixed length, N,  of the complex  input signal amplitudes A_x (T_i,0),  A_y (T_i,0) ,i=1…N, with T_i-T_(i-1)= W/N, where W is the time window. The input signal is propagated along the fiber length in discrete, adaptive steps of length h_j, j=1,M to yield an output signal A_x (T_i,L), A_y (T_i,L), where L is the length of the optical fiber.

In the Symmetrized Split-Step Fourier Method (SSSFM) equations (1a) and (1b) are written in the form:

where

and where \(\bar{A}_{x}\) and \(\bar{A}_{y}\) denote the complex conjugates of A_x and A_y, respectively.

Equations (15a) and (15b) have solutions

Thus, the complex field amplitudes, A_x and A_y may be propagated one step h_j along the longitudinal (z) direction along by operating on them with the operator, exp⁡[h_j (D_x+N_x )], exp⁡[h_j (D_y+N_y)] , respectively.

Equations (18a) and (18b) are approximated by:

The operators  \(\exp \left[h_{j} D_{x} / 2\right]\) and  \(\exp \left[h_{j} D_{y} / 2\right]\) are applied by Fourier transformation to the frequency domain, by subsequent multiplication of A_x (z, ω_i) and A_y (z, ω_i) by the complex numbers:

respectively, and where Ax (z, ωi) and Ay (z, ωi) are the Fast Fourier Transforms (FFTs) of Ax (z, Ti) and Ay (z, Ti), respectively, followed by inverse Fourier transformation back to the time domain.

The operators \(\exp \left[h_{j} \hat{N}_{y}\right]\) and \(\exp \left[h_{j} \hat{N}_{y}\right]\) are evaluated in the time domain and again correspond to multiplication by the complex numbers:

respectively, and where the time derivatives are approximated by central differences as follows:

Therefore, the method used to propagate the complex field amplitudes one step of length h_j is given by

where the expressions for  \(\exp \left[h_{j} D_{x} / 2\right]\) , \(\exp \left[h_{j} D_{y} / 2\right]\) ,\(\exp \left[h_{j} \hat{N}_{x}\right]\) and \(\exp \left[h_{j} \hat{N}_{y}\right]\) are as given in (20a, b) and (21a, b) above, respectively.

The adaptive step length is determined at the beginning of each step according to

[3] where P₀ is the maximum instantaneous input power of the waveform and \( \Phi_{MAX}^{NL} \) is the upper limit on the nonlinear phase increment per step, a numerical input parameter typically set to a value of  0.05.

Inclusion of Polarization Mode Dispersion

Polarization Mode Dispersion (PMD) arises out of small unintended imperfections in the optical fiber leading to a modification of the propagations indices of light polarized along the two orthogonal directions (fast and slow axes) in the transverse plane of the fiber. Moreover, the orientation of the fast and slow axes varies randomly along the longitudinal length of the fiber.

When PMD effects are included, the fiber is modeled in INTERCONNECT in what is known in the literature as the “Coarse Step Method”. This is accomplished by first dividing the fiber into a number of PMD sections, each of length \( l^{PMD}_k \), k=1…K such that \( \sum_{k=1}^{K} l_{k}^{P M D}=L \) and then propagating the complex field amplitudes through each such PMD section as described in Section 1 above, where the x and y axes correspond to the fast and slow principal axes, respectively, and where an additional PMD related differential group delay of Δβ'_PMD is added to the group delay specified which is calculated from the user supplied value of the PMD parameter, D_p (usually provided in manufacturer specifications sheets). It is assumed that the PMD parameter follows the mean convention (as opposed to the root-mean-square convention) such that

where the angle brackets denote the mean over all the observations, and \( \left\langle\Delta \tau_{f s}\right\rangle \) is the observed maximal time delay between pulses polarized along two orthogonal transverse directions.

The additional PMD related differential group delay in each PMD section is then calculated according to

As well, a random rotation of θ_k is applied to the principal axes and a phase of ϕ_k is applied to the values of the complex field amplitudes at the boundaries between the PMD sections as depicted in Figure 1. The random rotations, θ_k are drawn from a uniform distribution in the range 0 to 2π, and the random phases ϕ_k are drawn from the uniform distribution over the range -π/2 to π/2.

The lengths of the PMD step sizes, \( l^{PMD}_k \) are drawn from a Gaussian distribution with mean, \( \mu_{s t e p}^{P M D} \) , and standard deviation \( \sigma^{PMD}_{step} \) , which are specified by the user.

Table of Symbols and Quantities and INTERCONNECT parameters

Table of Symbols and Quantities and INTERCONNECT parameters

REFERENCES

[1] “Nonlinear Fiber Optics, 4^th ed.”, G. Agrawal, Elsevier Academic Press, 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA, 2007.

[2] Ralf Deiterding and Stephen W. Poole, “Robust split-step Fourier methods for simulating the propagation of ultra-short pulses in single and two-mode optical communication fibers”, arXiv:1504.01331v1 [math.NA] 6 Apr 2015.

[3] Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis R. Menyuk,, “Optimization of the Split-Step Fourier Method inModeling Optical-Fiber Communications Systems”, Journal Of Lightwave Technology, Vol. 21, No. 1, January 2003.