Models linear and nonlinear effects in waveguides (block mode)

## 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. |
NLSE Waveguide | - | - |

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). |
NLSE Waveguide | - | - |

A brief description of the elements functionality. |
Models linear and nonlinear effects in waveguides (block mode) | - | - |

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

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 |

Central frequency of the waveguide. A Taylor expansion around this frequency is performed to estimate the propagation transfer function of the waveguide. |
1552.524381 | nm* *std. unit is Hz |
(2.99792e-83, +∞) |

The length of the waveguide. |
1 | km* *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 | - | - |

The loss corresponding to the first orthogonal identifier. |
0 | dB/m | [0, +∞) |

The group index coefficient corresponding to the first orthogonal identifier. |
0 | - | [0, +∞) |

The dispersion coefficient corresponding to the first orthogonal identifier. |
16e-006 | s/m/m | (-∞, +∞) |

Defines the dispersion slope corresponding to the first orthogonal identifier. |
0 | s/m^2/m | (-∞, +∞) |

Defines the nonlinear coefficient of the first orthogonal identifier. |
124 | 1/m/W | (-∞, +∞) |

Defines the influence of the generated free carriers of mode 1 on the nonlinear index and waveguide birefringence. |
1 | - | (-∞, +∞) |

### Waveguide/Mode 2 Properties

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

Defines whether or not to enable coupled mode calculation. |
false | - | [true, false] |

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 second orthogonal identifier. |
Y | - | - |

The loss corresponding to the second orthogonal identifier. |
0 | dB/m | [0, +∞) |

The group index coefficient corresponding to the second orthogonal identifier. |
0 | - | [0, +∞) |

The dispersion coefficient corresponding to the second orthogonal identifier. |
16e-006 | s/m/m | (-∞, +∞) |

Defines the dispersion slope corresponding to the second orthogonal identifier. |
0 | s/m^2/m | (-∞, +∞) |

Defines the nonlinear coefficient of the second orthogonal identifier. |
50.5 | 1/m/W | (-∞, +∞) |

Defines the nonlinear coupled coefficient of the first and second orthogonal identifiers. |
67.5 | 1/m/W | (-∞, +∞) |

Defines the influence of the generated free carriers of mode 2 on the nonlinear index and waveguide birefringence. |
1 | - | (-∞, +∞) |

### Waveguide/Nonlinearities Properties

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

Defines whether or not to include nonlinear effects. |
true | - | [true, false] |

Defines whether or not to include self-steepening effects. |
true | - | [true, false] |

Defines whether the nonlinear coefficient values are defined as a table with frequency dependent values or calculated from the nonlinear index and effective area. |
parameters | - | [parameters, table |

Defines the nonlinear refractive index. |
6e-018 | m^2/W | (-∞, +∞) |

Defines the fiber's effective core area. |
80e-012 | m^2 | [0, +∞) |

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

The file containing the frequency dependent nonlinear coefficient values. Refer to the Implementation Details section for the format expected. |
- | - | - |

The table containing the frequency dependent nonlinear coefficient values |
<2> [193.1e+012, 0] | - | - |

### Waveguide/Nonlinearities/Raman Scattering Properties

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

Defines whether or not to include nonlinear Raman effects. |
true | - | [true, false] |

Defines how Raman parameters are specified. |
constant | - | [constant, impulse response, spectrum |

Defines the Raman time constant; |
3 | fs* *std. unit is s |
[0, +∞) |

Defines the Raman impulse response fractional contribution. |
0.18 | ratio | [0, 1] |

Defines the first Raman impulse response time constant |
12.2 | fs* *std. unit is s |
(0, +∞) |

Defines the second Raman impulse response time constant |
32 | fs* *std. unit is s |
(0, +∞) |

### Numerical Properties

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

Defines if noise bins are incorporated into the signal waveform. |
true | - | [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, +∞) |

### Numerical/Digital Filter Properties

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

Defines whether or not to use a single tap digital filter to represent the element transfer function in time domain. |
false | - | [true, false] |

Defines the method used to estimate the number of taps of the digital filter. |
fit tolerance | - | [disabled, fit tolerance, group delay |

Defines the mean square error for the fitting function. |
0.001 | - | (0, 1) |

Defines the window type for the digital filter. |
rectangular | - | [rectangular, hamming, hanning |

Defines the number of coefficients for digital filter. |
256 | - | [1, +∞) |

Defines the number of coefficients for digital filter. |
4096 | - | [1, +∞) |

Defines the time delay equivalent to a number of coefficients for digital filter. |
0 | s | [0, +∞) |

Defines whether to use the initial input signal to initialize filter state values or to set them to zero values. |
false | - | [true, false] |

### Diagnostic Properties

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

Enables the frequency response of the designed filter implementation and the ideal frequency response to be generated as results. |
false | - | [true, false] |

The number of frequency points used when calculating the filter frequency response. |
1024 | - | [2, +∞) |

### Waveguide/Nonlinearities/Two-photon absorption and Free carriers Properties

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

Defines whether or not to include two-photon absorption effects. |
true | - | [true, false] |

Defines the two-photon absorption coefficient. |
5e-012 | m/W | [0, +∞) |

Defines whether or not to include free carrier effects. |
true | - | [true, false] |

Free carrier induced nonlinearities time constant. |
1e-009 | s | [0, +∞) |

Defines the free carrier absorption cross-section that accounts for FCA effect. |
1.45e-021 | m^2 | [0, +∞) |

Defines how the free carrier dispersion is calculated. |
effective | - | [effective, electron-hole |

Defines the effective free carrier dispersion coefficient that accounts for FCD effect. |
1.35e-027 | m^3 | [0, +∞) |

Defines the electron free carrier dispersion coefficient that accounts for FCD effect. |
0.88e-027 | m^3 | [0, +∞) |

Defines the hole free carrier dispersion coefficient that accounts for FCD effect. |
0.46e-027 | m^3 | [0, +∞) |

## Results

Name | Description |
---|---|

diagnostic/response #/transmission | The complex transmission vs. frequency corresponding to the ideal and designed filter. |

diagnostic/response #/gain | The gain vs. frequency corresponding to the ideal and designed filter. |

diagnostic/response #/error | Mean square error comparing the frequency response of the designed filter implementation with the ideal frequency response. |

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

## Implementation Details

### Overview

The propagation of arbitrarily polarized light in a waveguide is governed by the following pair of coupled generalized nonlinear Schrödinger equations given in equation (1) [1]:

$$ \begin{gathered} \frac{\partial A_{1}}{\partial z}=\left(\sum_{n=2}^{\infty} i^{n+1} \frac{\beta_{\mathrm{n}}^{(1)}}{n !} \frac{\partial^{n}}{\partial T^{n}}\right) A_{1}-\frac{\alpha_{1}}{2} A_{1} \\ +i \gamma_{11}\left((1+i r)\left|A_{1}\right|^{2}+\frac{i}{\omega_{0}} \bar{A}_{1} \frac{\partial}{\partial T} A_{1}+\left[\frac{i}{\omega_{0}}-T_{R} \frac{\partial}{\partial T}\right]\left|A_{1}\right|^{2}\right) A_{1}+\gamma_{12}(1+i r)\left|A_{2}\right|^{2} A_{1} \\ -\left(\frac{\sigma}{2} N_{c}-i \frac{\omega_{0}}{c} n_{f}\right) A_{1} \end{gathered} $$ |
(1a) |

$$ \begin{aligned} \frac{\partial A_{2}}{\partial z}=&\left(\sum_{n=1}^{\infty} i^{n+1} \frac{\beta_{\mathrm{n}}^{(2)}}{n !} \frac{\partial^{n}}{\partial T^{n}}\right) A_{y}-\frac{\alpha_{2}}{2} A_{2} \\ +i \gamma_{22}\left((1+i r)\left|A_{2}\right|^{2}+\frac{i}{\omega_{0}}\right.&\left.\bar{A}_{2} \frac{\partial}{\partial T} A_{2}+\left[\frac{i}{\omega_{0}}-T_{R} \frac{\partial}{\partial T}\right]\left|A_{2}\right|^{2}\right) A_{2}+\gamma_{12}(1+i r)\left|A_{1}\right|^{2} A_{2} \\ &-\left(\frac{\sigma}{2} N_{c}-i \frac{\omega_{0}}{c} n_{f}\right) A_{2} \end{aligned} $$ |
(1b) |

where:

\(A_1\) and \(A_2\) 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,

\(z\) is the longitudinal spatial coordinate,

\(\omega_0=2\pi f_0\) is the central angular frequency of the simulation band,

\(\alpha_1\) and \(\alpha_2\) are the linear absorption coefficients for mode 1 and 2, respectively,

\(T\) is the retarded time given by:

$$ T=t-\Delta \beta_{1} Z $$ |
(2) |

where \(t\) is the time coordinate, and \(\Delta\beta_1\) is the differential group delay and can be expressed in terms of the group index as:

$$ \Delta \beta_{1}=\frac{\beta_{1}^{(1)}-\beta_{2}^{(1)}}{2}=\frac{n_{g 1}-n_{g 2}}{2 c}=\frac{\Delta n_{g}}{2 c} $$ |
(3) |

and \(\beta_2\) and \(\beta_3\) are given by:

$$ \beta_{2}=\left.\frac{\partial^{2}}{d \omega^{2}} \beta(\omega)\right|_{\omega=\omega_{0}} $$ |
(4) |

$$ \beta_{3}=\left.\frac{\partial^{3}}{d \omega^{3}} \beta(\omega)\right|_{\omega=\omega_{0}} $$ |
(5) |

Commonly values for dispersion, \(D\) and dispersion slope, \(D'\) , are reported, and \(\beta_2\) and \(\beta_3\) are related to them by the following relations:

$$ \beta_{2}=\frac{-2 \pi c}{\omega_{0}^{2}} D $$ |
(6) |

$$ \beta_{3}=\frac{1}{\omega_{0}^{2}}\left(\frac{4 \pi c}{\omega_{0}} D+\frac{4 \pi^{2} c^{2}}{\omega_{0}^{2}} D^{\prime}\right) $$ |
(7) |

\(\gamma_{11}\) and \(\gamma_{22}\) are the nonlinear parameter for mode 1 and 2, respectively,

\(\gamma_{12}\) is the coupled nonlinear parameter,

\(r\) is given by:

$$ r=\frac{\beta_{T P A} C}{2 \omega_{0} n_{2}} $$ |
(8) |

Where \(\beta_{TPA}\) is the two-photon absorption (TPA) coefficient, \(c\) is the speed of light, and \(n_2\) is the nonlinear index.

The Raman time constant, \(T_R\), can be calculated from

$$ T_{R}=f_{R} \int_{0}^{\infty} t h_{R}(t) d t $$ |
(9) |

where \(f_R\) is the Raman response fractional contribution, and \(h_R(t)\) 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

$$ h_{R}(t)=\frac{\tau_{1}^{2}+\tau_{2}^{2}}{\tau_{1} \tau_{2}^{2}} \exp \left(-\frac{t}{\tau_{2}}\right) \sin \left(\frac{t}{\tau_{1}}\right) $$ |
(10) |

where \(\tau_{1}\) is the Raman response time constant 1, and \(\tau_{2}\) is the Raman response time constant 2. The integral in equation (9) can be performed analytically, or approximated as [2]:

$$ T_{R}=f_{R} \frac{\tau_{1} \tau_{2}^{2}}{\tau_{1}^{2}+\tau_{2}^{2}} $$ |
(11) |

\(\sigma\) governs the free-carrier absorption (FCA), and \(n_f\) is the free-carrier index (FCI) which is given by:

$$ n_{f}=-N_{c} \sigma_{n}^{e f f} $$ |
(12) |

Alternatively, FCI can be expressed in terms of the electron and hole contributions as:

$$ n_{f}=-N_{c} \sigma_{n}^{e}-\left(N_{c} \sigma_{n}^{h}\right)^{0.8} $$ |
(13) |

Finally, \(N_c\) is the carrier density, and a rate equation is required to describe the dynamics of carrier density:

$$ \frac{d N}{d t}=G-\frac{N}{\tau_{f c}} $$ |
(14) |

where \(\tau_{f c}\) is the carrier density effective life time that includes all the effects of recombination, diffusion and drift. A standard value of 1ns is used for silicon waveguides. \(G\) is the generation rate and is given by:

$$ G=\frac{r}{h f_{0} A}\left(\gamma_{11}\left|A_{1}\right|^{4}+\gamma_{22}\left|A_{2}\right|^{4}+2 \gamma_{12}\left|A_{1} A_{2}\right|^{2}\right) $$ |
(15) |

where \(h\) is Plank’s constant, and \(A\) is the cross-section area of the waveguide.

If ‘coupled mode’ option is disabled, then the nonlinear coefficient, \(\gamma\) , is calculated as

$$ \gamma\left(\omega_{0}\right)=\frac{n_{2}\left(\omega_{0}\right) \omega_{0}}{c A} $$ |
(16) |

and the nonlinear coefficients in equations (1) and (15) are set as:

$$\gamma_{11}= \gamma,$$

$$\gamma_{22} = 0,$$

$$\gamma_{12} = 0$$

In INTERCONNECT, the calculation is performed in block mode by discretizing and integrating equations (1a) and (1b) along with the rate equation (15) using the Symmetrized Split-Step Fourier Method (SSSFM) [2].

### Example

In this section, we present two examples and compare with the literature for both coupled mode and single mode propagation.

In the file [[NLSE_WGD_Yin2007.icp]], the source is a 16.7 ps Gaussian pulse. The effect of self-phase modulation (SPM), two-photon absorption (TPA) and free carriers (FC) are presented. As expected, SPM causes spectral broadening for an unchirped Gaussian pulses. The TPA affects the pulse amplitude due to the absorption but also narrows the spectral width of the pulse. The FC induced by TPA makes the pulse asymmetric, affecting the red part of the spectrum, as can by seen in the figure above.

In the file [[NLSE_WGD_Daniel2010_Fig9.icp]], the source is a 15 ps Gaussian pulse that is linearly polarized at 45 degrees. The ‘coupled mode’ is enabled and all nonlinear effects (SPM, TPA and FC) are enabled except Raman. The stokes parameters \(S0\), and \(S1\) are shown below in comparison with Fig.9 in ref[1]. As can be seen from S1, there is a temporal region where the polarization state switches from 1 (45 degrees) to -1 (-45 degrees), due to the temporal changes in the birefringence induced by free carriers.

### Related Publications

[1] Brian A. Daniel and Govind P. Agrawal, "Vectorial nonlinear propagation in silicon nanowire waveguides: polarization effects," J. Opt. Soc. Am. B 27, 956-965 (2010)

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

[3] L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007).