Optical fiber is widely used in long-haul communication systems as a transmission media due to its low attenuation and very high transmission bandwidth. Fiber optic cables are also immune to problems like electromagnetic interference and the light signal in the fiber can be easily amplified in the transmission link.

However, given the nature of the light and that optical fiber is a dispersive media, light signal transmitted in optical fibers suffers from dispersion problems. In this section, we demonstrate the pulse broadening effects in optical fibers due to linear dispersion (Gaussian Pulse Broadening in the Linear Regime) and the combination of linear and nonlinear dispersion including group velocity dispersion (GVD), self-phase modulation and self steepening (Gaussian Pulse Propagation in nonlinear Regime), respectively.

## Gaussian Pulse Broadening in the Linear Regime

This example demonstrates the propagation of a Gaussian pulse in the linear dispersion regime of a fiber. Due to a phenomenon known as Group Velocity Dispersion, as an optical pulse with a Gaussian temporal profile travels down an optical fiber operating in the linear regime it maintains its Gaussian temporal profile but the width of the Gaussian broadens.

Due to a phenomenon known as Group Velocity Dispersion, as an optical pulse with a Gaussian temporal profile travels down an optical fiber operating in the linear regime it maintains its Gaussian temporal profile but the width of the Gaussian broadens according to:

$$ T_{1}(z)=T_{0}\left[1+\left(\frac{z}{L_{D}}\right)^{2}\right]^{\frac{1}{2}} $$ |
(1) |

where:

z is the distance traveled along the fiber;

T_{0} is the half-width at the 1/e intensity point of the input pulse (at z=0) which is related to the full width at half maximum, T_{FWHM} by:

$$ T_{0}=\frac{T_{F W H M}}{2[\ln (2)]^{\frac{1}{2}}} $$ |
(2) |

T_{1} (z) is the half-width at the 1/e intensity point after having propagated a distance z;

L_{D} is known as the dispersion length and is given by:

$$ L_{D}=\frac{2 \pi c T_{0}^{2}}{\lambda^{2} D} $$ |
(3) |

where c is the speed of light, λ is the (center) wavelength of the light and D is the dispersion parameter which is given by:

$$ D=-\frac{\lambda}{c} \frac{d^{2} n_{e f f}}{d \lambda^{2}} $$ |
(4) |

where n_{eff} is the effective index of the optical mode of the fiber.

The value of the dispersion parameter is usually provided in manufacturer specification sheets.

The spreading of the pulse in the time domain can be understood by considering the fact that due to the phenomenon of Group Velocity Dispersion the different frequency component of the pulse will travel down the fiber at different speeds.

### Instruction

1) Open the INTERCONNECT Project File (.icp file) file named “AgrawalFig3_1.icp”.

2) Open the Lumerical Script File (.lsf file) named “calculateParmsFig3_1.lsf”.

3) Run the “calculateParmsFig3_1.lsf” script.

4) Run the simulation.

5) Open the Lumerical Script File (.lsf file) named “plotResultFig3_1.lsf”.

6) Run the “plotResultFig3_1.lsf” script.

### Results

The circuit in the example file is:

The scripts will produce a plot with the following traces:

a) the input pulse shape to the fiber (normalized to 1), as generated by the pulse generator “GAUSS_1” and measured by “OOSC_1”. Obviously, this corresponds to shape of the pulse after having traveled a distance of 0 down the fiber.

b) the shape of the pulse, as calculated by INTERCONNECT, after having traveled down a distance of z/L_{d} =2.

c) the shape of the pulse, as presented in Figure 3.1 of reference [1], after having traveled down a distance of z/L_{d} =2.

## Gaussian Pulse Propagation in Nonlinear Regime

This example demonstrates the propagation of a Gaussian pulse in the normal dispersion regime of a fiber, D

This is an example of the propagation of a pulse in the normal dispersion regime of a fiber, D

The calculation requires a numerical solution to the Generalized Non-Linear Schrodinger Equation which is performed by INTERCONNECT using the Symmetrized Split-Step Fourier Method (SSSFM) [1,2].

In the example:

z is the distance traveled along the fiber;

T_{0} is the half-width at the 1/e intensity point of the input Gaussian pulse (at z=0) which is related to the full width at half maximum, T_{FWHM} by:

$$ T_{0}=\frac{T_{F W H M}}{2[\ln (2)]^{\frac{1}{2}}} $$ |
(1) |

L_{D} is known as the dispersion length and is given by:

$$ L_{D}=\frac{2 \pi c T_{0}^{2}}{\lambda^{2} D} $$ |
(2) |

where c is the speed of light, λ is the (center) wavelength of the light and D is the dispersion parameter which is given by:

$$ D=-\frac{\lambda}{c} \frac{d^{2} n_{e f f}}{d \lambda^{2}} $$ |
(3) |

and n_{eff} is the effective index of the optical mode of the fiber.

The value of the dispersion parameter is usually provided in manufacturer specification sheets.

L_{NL} is known as the nonlinear length and is given by:

$$ L_{N L}=\frac{1}{\gamma\left(\omega_{0}\right) P_{0}} $$ |
(4) |

where P_{0} is the input power of the pulse;

and ω_{0} is the angular frequency of the light and related to λ by:

$$ a_{b}=\frac{2 \pi c}{\lambda} $$ |
(5) |

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

$$ \gamma\left(\omega_{0}\right)=\frac{n_{2}\left(\omega_{0}\right) \omega_{0}}{c A_{e f f}} $$ |
(6) |

The effective area, Aeff, and nonlinear index, n_{2} (ω_{0}), are usually provided by optical fiber manufacturers in their data sheets.

N is known as the soliton number as is given by:

$$ N = \frac{L_D}{L_{NL}} $$ |
(7) |

s is known as the self-steepening parameter and is given by:

$$ s=\frac{1}{\omega_{0} T_{0}}\ $$ |
(8) |

### Instructions

1) Open the INTERCONNECT Project File (.icp file) file named “AgrawalFig4_21a.icp”.

2) Open the Lumerical Script File (.lsf file) named “calculateParmsFig4_21a.lsf”.

3) Run the “calculateParmsFig4_21.lsf” script.

4) Run the simulation.

5) Open the Lumerical Script File (.lsf file) named “plotResultFig4_21.lsf”.

6) Run the “plotResultFig4_21.lsf” script.

### Results

The parameters in this example are s=0.1, N=10. The circuit in the example file is shown below:

The scripts will produce a plot with the following traces:

a) the shape of the pulse, as calculated by INTERCONENCT, after having traveled down the fiber a distance of z/L_{d} =0.2.

b) the shape of the pulse, as presented in Figure 4.21(a) of reference [1], after having traveled down the a distance of z/L_{d} =0.2.

## Single Mode Fiber-28 (SMF-28) example

In this section, we demonstrate an optical fiber transmission link with a piece of normal Single Mode Fiber (SMF28) for Quadrature Phase Shift Keying (QPSK) modulation.

Following is the circuit in the example file smf28_qpsk.icp.

The property setting for the fiber named "FIBER" is listed in the table below and the setting is based on the Corning SMF28 fiber:

reference frequency |
193.05 THz |

length |
100 km |

attenuation |
0.2 dB/km |

dispersion |
-18 ps/nm/km |

dispersion slope |
0 ps/nm^2/km |

nonlinear index |
2.6e-20 m^2/W |

effective area |
7.85398e-11 m^2 |

raman gain time constant |
3 fs |

differential group delay |
0 s/m |

pmd coefficient |
3.16228e-15 s/m^.5 |

The fiber "DISPCOMP" is a piece of linear Dispersion Compensation Fiber (DCF) and it compensates the linear dispersion in the fiber "FIBER". The amplifier "AMP" has a gain of 20 dB and it compensates the attenuation in the fiber for exactly 100 km length. The transmission fiber (100 km), dispersion compensation fiber and the amplifier forms one span of the transmission channel and the Loop Control element controls the number of spans for the whole transmission link.

For comparison, we plot the constellation diagram of the signal at the receiver end for four distortion conditions in the fiber: no dispersion, linear dispersion only, nonlinear dispersion only and linear + nonlinear dispersion for 1 span and 8 spans, respectively.

no dispersion

linear dispersion only

nonlinear dispersion only

linear + nonlinear dispersion

QPSK constellation diagram with 1 span of fiber transmission link

no dispersion

linear dispersion only

nonlinear dispersion only

linear + nonlinear dispersion

QPSK constellation diagram with 8 span of fiber transmission link

### References

1. “Nonlinear Fiber Optics, 4th 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

### See also

Optical Fiber INTERCONNECT element