This page provides the theory behind modulation response calculations.

## Overview

Many of the key performance metrics of an electro-optic modulator can be extracted from the modulation response, including VπLπ, the insertion loss, and the extinction ratio. In this document, we will describe the procedure for determining the modulation response using the effective index of the waveguide. The effective index can be determined as a function of applied bias using a combination of electrical and optical simulation of the component. For the purposes of this example, a simple Mach-Zehnder modulator (MZM) will be used. The physical layout of a typical MZM is shown below. It consists of an input and output Y-branch and two arms. By applying a voltage to e.g. a PN diode on one arm, the effective index of the waveguide can be changed, altering the effective length of the arm along which the optical wave will travel. This change in effective length alters the phase accumulated by the wave traveling on the active arm, which will then interfere with the unperturbed wave at the output Y-branch.

A simple schematic view of the component is shown in the figure below. The active arm is represented by a measured optical modulator element.

## Calculating the Transmission

The optical signal at the output can be described as the sum of two waves,

$$E\left(V_{1}, V_{2}\right)=\frac{E_{0}}{1+\sigma}\left[\sigma \exp \left(-\frac{2 \pi}{\lambda_{0}} n_{eff}\left(V_{1}\right) L\right)+\exp \left(-\frac{2 \pi}{\lambda_{0}} n_{e f f}\left(V_{2}\right) L\right)\right]$$

where σ represents the splitting ratio (the ratio of the power in each branch), L is the length of the arm, λ0 is the free-space wavelength, and neff(V) is the effective index of the arm as a function of the applied voltage.

Note that neff(V) is complex. In the case where only one arm is actively driven,

$$n_{e f f}\left(V_{2}\right)=n_{e f f,2} $$

and the phase accumulated in that arm is a constant

$$\varphi_{2}=\frac{2 \pi}{\lambda_{0}} n_{e f f,2}L$$

The normalized power transmission can then be calculated as

$$T\left(V_{1}\right)=\left|\frac{E\left(V_{1}\right)}{E_{0}}\right|^{2}=\left|\frac{1}{1+\sigma}\left[\sigma \exp \left(-\frac{2 \pi}{\lambda_{0}} n_{e f f}\left(V_{1}\right) L\right)+\exp \left(-\varphi_{2}\right)\right]\right|^{2}$$

## Modulator Metrics

## V\(_{\pi}\)L\(_{\pi}\)

The VπLπ product can be determined from the effective index resulting in change in phase of π:

$$\Delta \varphi=\frac{\Delta n_{e f f} 2 \pi L}{\lambda_{0}} \rightarrow \Delta n_{e f f}\left(V_{pi}\right)=\frac{\lambda_{0}}{2 L_{pi}}$$

By inverting the relationship between neff and V. For a given L, Vπ can also be read from the transmission plot as the location of the minimum.

## Insertion Loss

The insertion loss will be the ratio of the peak normalized transmission and the ideal transmission (1). The quantity is reported in dB,

$$I L=-10 \log _{10}(\max T)$$

Typical values will range from 0.1-2dB.

## Extinction Ratio

The extinction ratio will be the ratio of the peak transmission power to the minimum transmission power, and is reported in dB:

$$E R=10 \log _{10}\left(\frac{\max T}{\min T}\right)$$

Typical values are near 20dB.

## Example

Open and run the provided script file, mod_response.lsf. In this script, for a range of voltages from 0.5-1 volts, a range of corresponding effective indices have been defined. Note that this is just an example script to calculate the modulation response using the above equations and it assumes the voltage and index values are already defined. For a more comprehensive example of a voltage dependant material index, visit the PIN Mach Zehnder example.

Assuming a modulation length of 1mm:

Vπ, insertion loss and extinction ratio are then calculated to be :

Parameter |
Value (Approx.) |
---|---|

Vπ |
0.95 V |

Extinction ratio |
20 dB |

Insertion loss |
0.04 dB |