modulates an optical signal depending on electrical signal

## Keywords

electrical, optical, bidirectional

## Ports

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

port 1 | Optical Signal |

modulation | Electrical Signal |

port 2 | Optical Signal |

## Properties

### General Properties

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

Defines the name of the element. |
Traveling Wave Optical Modulator Measured | - | - |

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). |
Traveling Wave Optical Modulator Measured | - | - |

A brief description of the elements functionality. |
modulates an optical signal depending on electrical signal | - | - |

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

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 operation. |
193.1 | THz* *std. unit is Hz |
(0, +∞) |

The interaction length of the modulator. |
1 | m | [0, +∞) |

Defines whether the input parameter is a table with voltage dependent values or coefficients of a polynomial function. |
table | - | [table, coefficients |

### Standard/Table Properties

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

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

The file containing the measurement data. Refer to the Implementation Details section for the format expected. |
- | - | - |

Defines the type of measurement data. |
effective index | - | [absorption & phase, effective index |

A matrix editor for users to read the element current modulation data values. |
<11,3> [-5, -4.5, -4,...] | - | - |

### Standard/Coefficients Properties

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

The polynomial coefficient for the absorption function. |
-0.0617 | dB/V^3/m | (-∞, +∞) |

The polynomial coefficient for the absorption function. |
-0.2804 | dB/V^2/m | (-∞, +∞) |

The polynomial coefficient for the absorption function. |
-0.6635 | dB/V/m | (-∞, +∞) |

The polynomial coefficient for the absorption function. |
-0.0719 | dB/m | (-∞, +∞) |

The polynomial coefficient for the phase function. |
-0.0038 | rad/V^3/m | (-∞, +∞) |

The polynomial coefficient for the phase function. |
0.1132 | rad/V^2/m | (-∞, +∞) |

The polynomial coefficient for the phase function. |
-0.4825 | rad/V/m | (-∞, +∞) |

The polynomial coefficient for the phase function. |
-0.0107 | rad/m | (-∞, +∞) |

### Waveguide Properties

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

List of optical mode labels supported by the element. |
TE,TM | - | - |

### Enhanced Properties

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

Central frequency of operation. |
10 | GHz* *std. unit is Hz |
(0, +∞) |

Defines the microwave loss. |
0 | dB/m | [0, +∞) |

Defines the the microwave group index. |
3 | - | [0, +∞) |

Defines the waveguide effective index. |
3 | - | [0, +∞) |

Defines the source resistance. |
50 | Ohms | [0, +∞) |

Defines the source reactance. |
0 | Ohms | (-∞, +∞) |

Defines the characteristic resistance. |
50 | Ohms | (0, +∞) |

Defines the characteristic reactance. |
0 | Ohms | (-∞, +∞) |

Defines the terminating resistance. |
50 | Ohms | [0, +∞) |

Defines the terminating reactance. |
0 | Ohms | (-∞, +∞) |

### Numerical Properties

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

Determines the size of the spatial grid spacing used during the microwave transmission line simulation, defined as a fraction of the Courant numerical stability limit (must be less than 1 for numerical stability). |
1 | - | (0, 1] |

### Diagnostic Properties

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

Enables the gain and spontaneous emission spectrum response to be generated as results. |
true | - | [true, false] |

Enables junction capacitance fitting coefficients. |
false | - | [true, false] |

Enables unloaded characteristic impedance profile as a function of frequency. |
false | - | [true, false] |

Enables output of the time-varying voltage at the load. |
true | - | [true, false] |

### Enhanced/Junction Properties

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

Defines the junction resistance. |
0 | ohm.m | [0, +∞) |

Defines if the junction capacitance is constant or voltage dependent. |
constant | - | [constant, table |

Defines the junction capacitance. |
0 | F/m | [0, +∞) |

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

The file containing the measurement data. Refer to the Implementation Details section for the format expected. |
- | - | - |

A matrix editor for users to read the element current data values. |
<30,2> [-3, -2.896551724, -2.793103448,...] | - | - |

The order of the polynomial used to fit the tabulated junction capacitance vs voltage data. |
2 | - | (2, 10] |

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

## Implementation Details

The Traveling Wave Optical Modulator Measured (TWOMM) element induces a phase shift through electro-optic (free-carrier interaction) effect. The electrodes are modeled as distributed transmission lines where the signal amplitude can vary in space and time. The TWOMM model is a time-domain model and is based on solving the transmission line differential equations:

$$ \begin{aligned} \frac{d V(x, t)}{d x} &=-R I(x, t)-L \frac{d I(x, t)}{d t} \\ \frac{d I(x, t)}{d x} &=-G V(x, t)-C \frac{d V(x, t)}{d t} \end{aligned} $$

The equivalent circuit of a section of a transmission line is shown in the figure below. Here \(v(x, t)\) and \(i(x, t)\) represent the instantaneous voltage and current at position \(x\) and time \(t\). The \(R\), \(L\), \(G\), and \(C\) represent the resistance, inductance, conductance, and capacitance per unit length, respectively. The reverse bias PN junction is modeled as a series distributed \(R_{j}\) and \(C_j\) elements in parallel with \(TL\).

The complex characteristic impedance, \(Z_0\) requires as an input to the model is for the de-embedded electrode only. This means it represents only the RGLC components excluding the diode branch.

The model extracts the RGLC component from the user provided complex \(Z_0\) and complex microwave refractive index (microwave loss and microwave index), according to [1]:

$$ \begin{gathered} Z_{0}=\sqrt{\frac{R+j 2 \pi f L}{G+j 2 \pi f C}} \\ \gamma=\sqrt{(R+j 2 \pi f L)(G+j 2 \pi f C)} \end{gathered} $$

where \(f\), is the input microwave frequency. If the combination of \(Z_0\) and \(\gamma\) is non-physical resulting in a negative value of any of the RLGC components, an error occurs, and simulation is aborted.

Different than the Optical Modulator Measured (OMM) element, the TWOMM can model the effect of the PN junction as a series of distributed resistances and junction capacitances. The junction capacitance can be a constant value or a function of voltage. If a table/file is used then the data is fit a polynomial of order that is specified by the user. The default value is 2 and the range is any integer from 2 to 10.

**Example**

In the following example, we test the implementation of a non-linear junction capacitance. In order to show how the TL voltage varies with non-linear junction capacitance, we add two tests with constant junction capacitance representing the upper and lower bound of the non-linear junction capacitance. Therefore, the three cases for the transmission line:

- \(Cj = C_{min}\)
- \(Cj = C_{max}\)
- Nonlinear \(Cj = [C_{min}, C_{max}]\)

The load voltage of test 3 is bounded by the first two test cases. The test circuit is shown below:

The script file [[TWOMM_LossyTL_nonlinearCj.lsf]] loads different junction capacitance for the three test cases to the TWOMM model, run the simulation and generates the optical output signal for comparison of the 3 cases:

**Frequency response for matched terminating impedance**

The frequency response of the TWOMM is represented by the S21 parameter. To linearize the transmission line response at a specific operating point, a small-signal analysis is performed and hence the S21 parameter is extracted. A test bench is created to sweep the frequency then calculates the S21 parameter. The parameters of the modulators are selected from the literature [2]. The frequency response is shown below, and the 3dB optical bandwidth is approximately 7 GHz which agrees with the analytical expression:

$$f_{3 d B}=\frac{1.9 c}{\pi l\left(n_{e}-n_{g}\right)}$$

where \(l\) is the length, \(n_e\) is the microwave index, \(n_g\) is the optical group index, and \(c\) is the speed of light.

## Related Publications

[1] David M. Pozar, “Microwave engineering”, 3rd Ed., Wiley (2005)

[2] Lin, S. H., and Shih-Yuan Wang. "High-throughput GaAs PIN electrooptic modulator with a 3-dB bandwidth of 9.6 GHz at 1.3 µm." Applied Optics 26.9 (1987).