Erbium doped fiber

## Keywords

optical, bidirectional

## Ports

Name | Type |
---|---|

port 1 | Optical Signal |

port 2 | Optical Signal |

## Properties

### General Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines the name of the element. |
Erbium Doped Fiber | - | - |

Defines whether or not to display annotations on the schematic editor. |
true | - | [true, false] |

Defines whether or not the element is enabled. |
true | - | [true, false] |

Defines the element unique type (read only). |
Erbium Doped Fiber | - | - |

A brief description of the elements functionality. |
Erbium doped fiber | - | - |

Defines the element name prefix. |
EDF | - | - |

Defines the element model name. |
- | - | - |

Defines the element location or source in the library (custom or design kit). |
- | - | - |

Defines the local path or working folder $LOCAL for the element. |
- | - | - |

An optional URL address pointing to the element online help. |
- | - | - |

### Standard Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines the bidirectional or unidirectional element configuration. |
bidirectional | - | [bidirectional, unidirectional |

The length of the waveguide. |
6 | m | [0, +∞) |

The Erbium ions in the doped fiber region. |
5e+024 | m^-3 | (0, +∞) |

The fluorescence lifetime of the Erbium metastable level. |
10 | ms*
*std. unit is s |
(0, +∞) |

The radius of the Erbium-doped fiber region. |
3.3 | um*
*std. unit is m |
(0, +∞) |

### Waveguide/Mode 1 Properties

Name | Default value | Default unit | Range |
---|---|---|---|

The first identifier used to track an orthogonal mode of an optical waveguide. For most waveguide, two orthogonal identifiers '1' and '2' are available (with the default labels 'TE' and 'TM' respectively). |
1 | - | [1, +∞) |

The label corresponding to the first orthogonal identifier. |
X | - | - |

### Waveguide/Mode 2 Properties

Name | Default value | Default unit | Range |
---|---|---|---|

The second identifier used to track an orthogonal mode of an optical waveguide. For most waveguide, two orthogonal identifiers '1' and '2' are available (with the default labels 'TE' and 'TM' respectively). |
2 | - | [1, +∞) |

The label corresponding to the first orthogonal identifier. |
Y | - | - |

### Waveguide/Cross Sections Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines whether or not to load wavelength dependent cross-section parameters from an input file or to use the currently stored values. |
false | - | [true, false] |

The file containing the wavelength dependent cross section parameters. Refer to the Implementation Details section for the format expected. |
- | - | - |

The table containing the wavelength dependent absorption and emission cross section parameters. |
<199,3> [0.9336e-006, 0.937218e-006, 0.940837e-006,...] | - | - |

### Waveguide/Confinement Factor Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines whether the confinement values are defined as a table with wavelength dependent values or calculated from fiber specifications. |
fiber | - | [fiber, table |

Defines the mode field model used to calculate the confinement factor from the fiber specifications. |
LP01 | - | [LP01, Marcuse, Petermann II, Myslinki |

The core radius of the fiber. |
3.3 | um*
*std. unit is m |
(0, +∞) |

The numerical aperture of the fiber. |
0.23 | - | (0, +∞) |

Defines whether or not to load wavelength dependent confinement factor values from an input file or to use the currently stored values. |
false | - | [true, false] |

The file containing the wavelength dependent confinement factor values. Refer to the Implementation Details section for the format expected. |
- | - | - |

The table containing the wavelength dependent confinement factor values. |
<2> [1.55e-006, 1] | - | - |

### Waveguide/Loss Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines whether the loss values are defined as a table with wavelength dependent values or a constant value. |
constant | - | [constant, table |

The background loss of the fiber. |
0 | dB/m | [0, +∞) |

Defines whether or not to load wavelength dependent loss from an input file or to use the currently stored values. |
false | - | [true, false] |

The file containing the wavelength dependent loss values. Refer to the Implementation Details section for the format expected. |
- | - | - |

The table containing the wavelength dependent loss values. |
<2> [1.55e-006, 0] | - | - |

### Waveguide/Concentration Quenching Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines the concentration quenching model. |
none | - | [none, homogeneous, inhomogeneous, combined |

The homogeneous model cooperative upconversion coefficient. |
1.1e-024 | m^3/s | [0, +∞) |

The inhomogeneous model relative number of clusters. |
0.2 | ratio | [0, 1] |

### Numerical Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines whether or not to add ASE noise to the output signal. |
true | - | [true, false] |

The center frequency of the generated noise spectrum. |
193.1 | THz*
*std. unit is Hz |
(0, +∞) |

The ASE noise spectral range. |
5 | THz*
*std. unit is Hz |
[0, +∞) |

Defines the noise bins frequency spacing. |
100 | GHz*
*std. unit is Hz |
(0, +∞) |

Defines if noise bins are incorporated into the signal waveform. |
false | - | [true, false] |

Defines whether or not to automatically create an unique seed value for each instance of this element. The seed will be the same for each simulation run. |
true | - | [true, false] |

The value of the seed for the random number generator. A value zero recreates an unique seed for each simulation run. |
1 | - | [0, +∞) |

This determines the convergence tolerance. |
0.001 | - | [0, +∞) |

This determines the maximum number of iterations required to reach converges. |
100 | - | [2, +∞) |

Defines the number of longitudinal sections. |
100 | - | [2, +∞) |

The minimum detectable signal power level. |
-60 | dBm*
*std. unit is W |
(-∞, +∞) |

Defines the relationship between Erbium concentration in cubic meters and in weight ppm units. |
10e+021 | ppm-wt | [0, +∞) |

### Diagnostic Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Enables running detailed analysis on the signal, pump and noise propagation (frequency and longitudinal dependencies). |
false | - | [true, false] |

The frequency upper value where a signal is considered a pump or not. |
1500 | nm*
*std. unit is Hz |
(2.99792e-83, +∞) |

### Diagnostic/Pumps Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines the plot format for the results. |
1D | - | [1D, 2D |

This option allow users to choose to plot in units of frequency or wavelength. |
wavelength | - | [frequency, wavelength |

Defines the power unit to plot the results. |
W | - | [W, dBm |

### Diagnostic/Signals Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines the plot format for the results. |
1D | - | [1D, 2D |

This option allow users to choose to plot in units of frequency or wavelength. |
frequency | - | [frequency, wavelength |

Defines the power unit to plot the results. |
dBm | - | [W, dBm |

### Diagnostic/Noise Properties

Name | Default value | Default unit | Range |
---|---|---|---|

Defines the plot format for the results. |
2D | - | [1D, 2D |

This option allow users to choose to plot in units of frequency or wavelength. |
frequency | - | [frequency, wavelength |

Defines the power unit to plot the results. |
dBm | - | [W, dBm |

====================================

## Implementation Details

### Introduction

Erbium-doped fiber amplifiers (EDFA) by far dominate as part of the backbone of long-haul optical fiber communications due to their low noise, high and broadband optical gain [1]. EDFA is an optical repeater device that is used to boost the intensity of optical signals being carried through a fiber optic communication system. A typical setup of a simple dual pump EDFA is shown case below:

The core of the EDFA is an optical fiber doped with the rare earth element erbium so that the glass fiber can absorb light at one frequency and emit light at another frequency.

The energy diagram of the Er^{3+} doped system is presented in the following figure. The pumping process takes place between the ground level ^{4}I_{15/2} and the excited level ^{4}I_{13/2} (1480 pump) or ^{4}I_{11/2} (980 pump) with respective fractional populations n_{1}, n_{2} and n_{3}.

The level ^{4}I_{11/2} has a very short lifetime 0.1-10 μs and relaxes nonradiatively to level ^{4}I_{13/2} and the laser transition takes place between the ^{4}I_{13/2} and ^{4}I_{15/2}. So, the behaviors of rare-earth-ion doped devices can be described in terms of two-level rate equations for the population inversion density (n_{2}), the pump field (P_{P}), the signal field (P_{S}), and the amplified spontaneous noise (P_{A}):

$$ n_{2} = \frac{\sum_{i} \frac{\sigma_{i}^{a} \Gamma_{i}^{P} P_{P,i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\sigma_{j}^{a} \Gamma_{j}^{P} P_{S, j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\sigma_{k}^{a} \Gamma_{k}^{A} P_{A k}^{\pm}}{h \nu_{k} A}}{\sum_{i} \frac{\left(\sigma_{i}^{a}+\sigma_{i}^{e}\right) \Gamma_{i}^{P} P_{P, i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\left(\sigma_{j}^{a}+\sigma_{j}^{e}\right) \Gamma_{j}^{P} P_{S, j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\left(\sigma_{k}^{a}+\sigma_{k}^{e}\right) \Gamma_{k}^{A} P_{A, k}^{\pm}}{h \nu_{k} A}+\frac{1}{\tau_{E r}}} $$ |
(1) |

$$ \frac{\partial P_{P, i}^{\pm}}{\partial z}=\pm \Gamma_{i}^{P} N_{E_{r}}\left(\sigma_{i}^{e} n_{2}-\sigma_{i}^{a} n_{1}\right) P_{P, i}^{\pm} \mp \alpha_{i} P_{P, i}^{\pm} $$ |
(2) |

$$ \frac{\partial P_{S, j}^{\pm}}{\partial z}=\pm \Gamma_{j}^{S} N_{E r}\left(\sigma_{j}^{e} n_{2}-\sigma_{j}^{a} n_{1}\right) P_{S, j}^{\pm} \mp \alpha_{j} P_{S, j}^{\pm} $$ |
(3) |

$$ \frac{\partial P_{A, k}^{\pm}}{\partial z}=\pm \Gamma_{k}^{A} N_{E r}\left(\sigma_{k}^{e} n_{2}-\sigma_{k}^{a} n_{1}\right) P_{A, k}^{\pm} \pm \Gamma_{k}^{A} 2 h \nu_{k} \Delta\nu_{k} \sigma_{k}^{e} N_{Er} n_{2} \mp \alpha_{i} P_{P, i}^{\pm} $$ |
(4) |

$$ \Gamma_{i}=\frac{\int_{s} \rho_{Er}(s) \Psi_{i}(s) d s}{\int_{s} \rho_{E r}(s) d s}, 1=n_{1}+n_{2} $$ |
(5) |

where the Γ are the overlap factors between the light-field modes and the erbium distribution, which also knows as the cross section; σ^{a} and σ^{e} are the absorption and emission cross sections, respectively; A is the effective area of the erbium distribution; ± represent the forward-travelling and backward-travelling directions. The high gain erbium-doped fiber amplifiers require high erbium concentrations (the typical concentration is 0.7*10^{-19} cm^{-3}), which breaks the assumption of isolated erbium ions and induces the dissipative ion-ion interaction via the energy transfer between neighboring erbium ions, resulting in a reduction in the pump efficiency. This quenching effect was added into the amplifier model via the homogeneous up-conversion process and pair-induced quenching (PIQ) [2]. The homogeneous model assumes that the ions are evenly distributed and the population inversion is given by [3].

$$ n_{2}=\frac{\sum_{i} \frac{\sigma_{i}^{a} \Gamma_{i}^{P} P_{P, i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\sigma_{j}^{a} \Gamma_{j}^{P} P_{S, j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\sigma_{k}^{a} \Gamma_{k}^{A} P_{A, k}^{\pm}}{h \nu_{k} A}}{ \sum_i \frac{\left(\sigma_{i}^{a}+\sigma_{i}^{e}\right) \Gamma_{i}^{P} P_{P, i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\left(\sigma_{j}^{a}+\sigma_{j}^{e}\right) \Gamma_{j}^{P} P_{S, j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\left(\sigma_{k}^{a}+\sigma_{k}^{e}\right) \Gamma_{k}^{A} P_{A, k}^{\pm}}{h \nu_{k} A}+\frac{1}{\tau_{E r}}+C_{u p} n_{2} N_{t}} $$ |
(6) |

where C_{up} is the concentration-independent and host-dependent two-particle up-conversion constant measured in m^{3}/s.

In the inhomogeneous model [4], it is assumed that the ions are not evenly distributed and there are two distinct species: clustered ions and single ions that cannot interact with each other. The total population inversion is the sum of the average population inversion of clustered ions, n_{2}^{p}, and the average population inversion of single ions, n_{2}^{SI}, given by

$$ n_{2} = n_{2}^{p} + n_{2}^{SI} $$ |
(7) |

with

$$ n_{2}^{p}=\frac{2 R \Psi}{\sum_{i} \frac{\left(\sigma_{i}^{a}+\sigma_{i}^{e}\right) \Gamma_{i}^{P} P_{P, i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\left(\sigma_{j}^{a}+\sigma_{j}^{e}\right) \Gamma_{j}^{P} P_{S, j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\left(\sigma_{k}^{a}+\sigma_{k}^{e}\right) \Gamma_{k}^{A} P_{A k}^{\pm}}{h \nu_{k} A}+\frac{1}{\tau_{E r}}} $$ |
(8) |

$$ n_{2}^{\mathrm{SI}}=\frac{(1-2 R) \Psi}{\sum_{i} \frac{\left(\sigma_{i}^{a}+\sigma_{i}^{e}\right) \Gamma_{i}^{P} P_{P, i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\left(\sigma_{j}^{a}+\sigma_{j}^{e}\right) \Gamma_{j}^{P} P_{S,j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\left(\sigma_{k}^{a}+\sigma_{k}^{e}\right) \Gamma_{k}^{A} P_{A, k}^{\pm}}{h \nu_{k} A}+\frac{1}{\tau_{Er}}} $$ |
(9) |

$$ \Psi=\sum_{i} \frac{\sigma_{i}^{a} \Gamma_{i}^{P} P_{P, i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\sigma_{j}^{a} \Gamma_{j}^{P} P_{S, j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\sigma_{k}^{a} \Gamma_{k}^{A} P_{A, k}^{\pm}}{h \nu_{k} A} $$ |
(10) |

where R is the relative number of clusters.

In fact, the interaction between erbium ions should include cooperative up-conversion and pair-ion quenching and the degradation of gain performance caused by concentration quenching should arise from two contributions: cooperative up-conversion between single ions and cluster-induced quenching. In the combined model, the total population inversion is the sum of the average population inversion of clustered ions, n_{2}^{p}, and the average population inversion of single ions, n_{2}^{SC}, given by

$$ n_{2} = n_{2}^{p} + n_{2}^{SC} $$ |
(11) |

with

$$ n_{2}^{s c}=\frac{(1-2 R) \Psi}{\sum_{i} \frac{\left(\sigma_{i}^{a}+\sigma_{i}^{e}\right) \Gamma_{i}^{P} P_{P, i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\left(\sigma_{j}^{a}+\sigma_{j}^{e}\right) \Gamma_{j}^P P_{P, j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\left(\sigma_{k}^{a}+\sigma_{k}^{e}\right) \Gamma_{k}^{A} P_{A, k}^{\pm}}{h \nu_{k} A}+\frac{1}{\tau_{E r}}+C_{u p}(1-2 R) n_{2}^{s c} N_{t}} $$ |
(12) |

$$ \Psi=\sum_{i} \frac{\sigma_{i}^{a} \Gamma_{i}^{p} P_{P, i}^{\pm}}{h \nu_{i} A}+\sum_{j} \frac{\sigma_{j}^{a} \Gamma_{j}^{p} P_{S,j}^{\pm}}{h \nu_{j} A}+\sum_{k} \frac{\sigma_{k}^{a} \Gamma_{k}^{A} P_{A, k}^{\pm}}{h \nu_{k} A} $$ |
(13) |

### Method

To solve coupled equations in the general scheme of bi-directional pumping, the computation involves a dual boundary value problem for the system of differential equations, as shown in the following figure. The numerical solution of the coupled differential equations can be done through using the Runge-Kutta methods and the relaxation method. The Runge-Kutta method involves solving the equations by propagating the light fields forward and backward along the fiber, using the boundary conditions for the signal, pump, forward amplified spontaneous emission (ASE), and backward ASE. The presence of counterpropagating backward ASE creates the necessity for this back-and-forth simulation. In addition, the relaxation method is used to converge to the desired precision.

### Reference

1. E. Desurvire, Erbium-doped fiber amplifiers: principles and applications, Wiley-Interscience, 2002.

2. Jiang, C., Hu, W. and Zeng, Q., Numerical analysis of concentration quenching model of Er 3+-doped phosphate fiber amplifier. IEEE journal of quantum electronics 39, 266-1271, 2003.

3. P. Blixt et al., “Concentration-dependent upconversion in Er -doped fiber amplifiers: Experiments and modeling,” IEEE Photon. Technol. Lett., vol. 3, p. 996, 1991.

4. E. Delevaque et al., “Modeling of pair-induced quenching in erbium doped silicate fibers,” IEEE Photon. Technol. Lett., vol. 5, pp. 73–75, Jan. 1993